Digital Design in Mechanized Tunneling

Abstract

Digital design methods are constantly improving the planning procedure in tunnel construction. This development includes the implementation of rule-based systems, concepts for cross-document and -model data integration, and new evaluation concepts that exploit the possibilities of digital design. For planning in tunnel construction and alignment selection, integrated planning environments are created, which help in decision-making through interactive use. By integrating room-ware products, such as touch tables and virtual reality devices, collaborative approaches are also considered, in which decision-makers can be directly involved in the planning process. In current tunneling practice and during planning stage, Finite Element (FE) simulations form an integral element in the planning and the design phase of mechanized tunneling projects. The generation of adequate computational models is often time consuming and requires data from many different sources. Incorporating Building Information Modeling (BIM) concepts offers opportunities to simplify this process by using geometrical BIM sub-models as a basis for structural analyses. In the following chapter, some modern possibilities of digital planning and evaluation of alignments in tunnel construction are explained in more detail. Furthermore, the conception and implementation of an interactive BIM and GIS integrated planning system, ‘‘BIM-to-FEM’’ technology which automatically extracts relevant information needed for FE simulations from BIM sub-models, the establishment of surrogate models for real-time predictions, as well as the evaluation and comparison of planning variants are presented.

6.1 Introduction

With the move toward digitalization in the construction industry, the rapidly developing field of information technology (IT) plays a dominant role in modern construction processes, in particular large projects like tunnel constructions. As a consequence, more and more digital project data is produced by several authorized project partners and is stored in diverse formats using different IT applications during different stages of the project. Additionally, the availability of computer-aided design (CAD) and sophisticated data management tools have also had noticeable impact on the way construction industries (including tunneling industries) record, store, retrieve, access, process, visualize, maintain, display, and manage important project data. However, in most cases the employed IT tools are standalone applications with limited inter-compatibility with other platforms because proprietary format for storing and accessing information are often different. A coherent data exchange between all applications and information sources within a project is for this reason not directly feasible in digital form without human intervention. As a result of this lack of compatibility, a high amount of errors occur during the exchange of information between applications and information is lost. Moreover, different information packages, from different project partners, which often describe the same object/element in the project, make it difficult to maintain the necessary level of integrity and consistency of project information.

In this regard, Building Information Modeling (BIM) has emerged to address these problems generated by decentralized data management. BIM presents an innovative and collaborative solution that allows for an organized and efficient workflow during the entire life cycle of the project: planning, design, construction, operation, maintenance and possible later demolition phases [25]. Although BIM methods were first developed to be applied to improve the organization of above-ground building projects, they have been, and are currently being extended to tunneling projects [11, 32, 43, 74], among other engineering infrastructure projects [39, 69, 85]. The capability of BIM to connect information databases to a graphical representation based on a parametric 3D models allows better visualization, coordination, and management of construction projects and accordingly, leads to a reduction of construction costs and errors [46].

The development of a real-time interactive digital platform for planning and design of tunneling projects, particularly in urban areas, provides a valuable tool to support the design process and enrich the decision-making. Such platform would allow for visual exploration of the whole project including parametric generation of feasible alignments considering all project design criteria [42, 79]. For this purpose, a digital interactive platform has been developed for collaborative planning process in tunneling projects, in which the numerical simulation, the simulation results, i.e., settlements, construction time, and the damage assessment of existing infrastructure, is integrated and visualized in real-time.

The proposed digital platform considerably reduces the user effort connected with the generation of simulation models. This platform supports data acquisition and model generation as well as the visualization of the important results of a numerical simulation in a 3D graphical model so as to be in line with Building Information Modeling (BIM) concepts. To this end, a fully automatic model generation assistant has been developed and integrated into the so-called TIM, “Tunneling Information Model” in order to enhance the usability and compatibility of a structural tunnel simulation with BIM methods. The automatic model generator is able to correctly model the geometry of the tunnel project, generate all relevant model components, extract geotechnical survey data stored in CAD format, discretize the FE mesh, and control the flow of the simulation.

6.2 Information Modeling

In the BIM-based approach, information is used as a basis for decision-making and project steering. Some information can only be processed through the exploration of constraints and dependencies. Through so-called interaction chains, relevant information can be dissected and structured. An interaction chain can be defined as a sequence of processes that build on each other and whose results can lead directly to the evaluation of relevant project constraints (see Fig. 6.1). The execution of interaction chains needs to be provided with relevant information. This includes existing product data as well as temporary simulation results. For the management of these information, a 4D-information model has been developed. In a tunneling project, information is usually time-dependent and is distributed on different systems and various data formats.

Fig. 6.1

Example of an interaction chain that considers settlement data and strains for risk classification

In accordance with the BIM concept for tunneling construction domains, the Tunnel Information Model (TIM) [42] establishes a holistic container for tunneling projects that also enables the integration of interaction chains for project evaluation. In the TIM data is integrated and linked by collecting, classifying, structuring and processing necessary information from the planning and construction phase. The structuring of multiple interacting documents has been realized by adopting the multi-model container concept [73]. Compared to existing approaches in the building sector lots of dynamic and time-dependent data must be stored. These include predictive models in the planning phase as well as real settlement data and machine data in the construction phase. A continuously adaption to the ground conditions based on up-to-date measurements and simulation data represents a major challenge. The basis of the TIM is build upon four essential product models, including a hybrid ground model, a boring machine model, a tunnel model and built environment model (Fig. 6.2). A detailed differentiation is provided in the following subsections.

Fig. 6.2

Tunnel information model consisting of the essential four submodels and time-dependent settlement and machine data [74]

6.2.1 Standards for Digital Design Tunneling

The digital design in construction requires standards for the exchange of information to be applied. In the context of the BIM method, a distinction between semantic information and geometric representation is made. While the geometry describes the visual representation of an element, the semantic provides descriptive information, such as model structure and object properties. The Industry Foundation Classes (IFC) standard [38] has been established as an open exchange format for BIM models. Traditionally, the IFC schema only supports building construction. However, in recent years, the context of IFC has increasingly being improved to support infrastructure domains in future. An proposal including bridges, rails, roads and port & waterways denoted as IFC4.3 is currently being reviewed by the International Standardization Organisation (ISO) to be the next major officially accepted version of the IFC. An extension for tunnels is also currently being developed by a project group of buildingSmart International [15]. The current version is proposed as IFC4.4, to which the SFB has substantially contributed to the requirements and data models for mechanized tunnels.

6.2.2 Hybrid Ground Model

The ground behavior is a fundamental basis for the overall construction process. A hybrid ground model has been developed in [33] to allow both a global and detailed consideration of the ground conditions. Based on borehole measurements, the profile of the ground can be depicted and transferred to a layer-based ground model. The geometry of these layers, in this case, is based on boundary representations (B-Rep). To support the analysis and assignment in a larger resolution, the layers are decomposed into voxel data (see Fig. 6.3). The detail of the voxel representation is decided upon the specific application. Algorithms, such as octree decomposition, are able to perform efficient decomposition and thus represent the ground model as discretized voxel models [47]. Regions of voxels are then combined based on their properties to represent layers of soil. High detail is required, e.g. when processing data close to the boring machine.

Both formats (B-Rep and voxels) are stored side-by-side, for which methods have been developed to support the integration, consistency and versioning [34]. The versioning also allows to update the ground conditions in the hybrid model and to use it for subsequent analyses. In the overall process, the hybrid ground data model makes it possible to incorporate different version (time-dependent), different resolutions (space) and different geometry representations (format). Throughout all versions, data can be requested and updated in both boundary representations and voxel representations.

Fig. 6.3

Concept for querying soil properties for a surface model (B-Rep) based on the hybrid ground data model

Based on the hybrid ground model, analyses and simulation models can efficiently request and process information. For example, a sophisticated octree-based soil model was used to calculate uncertainties based on the knowledge gained from the boreholes locations [47]. This investigation utilized kriging algorithms to map calculated uncertainty directly onto each voxel in the ground model. Several approaches for semivariogram models were tested to illustrate and make optimizations for strategic selection of preferences in projects (Fig. 6.4).

Fig. 6.4

Effect of change in applied semivariogram model for the highlighted voxel [47]

6.2.3 Tunnel Boring Machine Model

The tunnel boring machine has also been modeled and is included in the TIM collection of considered submodels, primarily because of the 4D planning process, that also requires the TBM to be included in planned projects. It is paricularly important to be able to locate the machine, the face of the tunnel and to monitor the progression of the excavation. Here, the focus is on Earth Pressure Balance Shield (EPB) boring machines. To store and exchange information regarding the machine, a proposal for the Industry Foundation Classes (IFC) model has been developed (Fig. 6.5). Therefore, a classification for the model elements has been developed by interaction between tunneling and informatics domains within the SFB. These include proper geometric descriptions as well as the assignment of necessary semantic information. In addition, the machine data were linked directly into the model so that they can be updated during tunnel driving and queried, for example, when performing a settlement analysis.

Fig. 6.5

Implemented IFC hierarchy for a TBM in a BIM based process

6.2.4 Tunnel Model

The tunnel model itself has also been realized using an IFC-based product model. The geometric representation of such a model can normally not be created by hand, because of the thousands of ring segments. Therefore, parametric modeling methods have been used to automatically create a model based on the cross-section template and the placement of the rings (Fig. 6.6). The placement can be considered as input in both ways, either from the planned positioning as well as from real positions of existing mounted rings. Considering alignment planning variants and settlement calculations, the position can be precisely defined. When calculating the positioning for a planned tunnel, the connectivity of the rings must be considered to calculate the exact position and orientation [57]. Each element is transformed locally to represent the ring rotation and general orientation, using rotation matrices ($M_{X}$, $M_{Y}$ and $M_{Z}$) along all axis and defined according to the precise position and direction for the ring on the alignment. Then, the transformed tunnel ring is moved on the position of the alignment ($M_{T}$), creating a tunnel model.

Fig. 6.6

Creating a parametric tunnel model, based on an alignment positioning and orientation

Storing the tunnel model as an IFC model data file must also consider resource consumptions. Therefore, an efficient modeling technique has been applied to directly generate a resource preserving model as well as allowing to optimize existing models [88]. When the geometry and properties of each ring or ring segment are clearly represented, the model for individual sections of a tunnel project can easily reach hundreds of megabytes. This affects the external storage in a file format and the performance of visualizations in subsequent analyses. Here, the STEP-format and the flyweight programming-pattern has been utilized to drastically reduce the resources required for modeling and data storage (Fig. 6.7). The optimization focuses on single ring segments ($A_{i}$) and compares whole tunnel rings. As the types of segments (i.e., an $A_{1}$ ring type of $k$ segments) and even complete ring designs ($A_{1}$-…-$A_{k}$) are highly recurring in a mechanized tunnel model, this can be exploited to share specific information amongst different occurrences. In computer science terms, that is called the flyweight design pattern, which enables the efficient reuse of recurring data. In a tunnel model, this affects both geometry representations as well as common attributes, which do not change among succeeding ring constructions and can be stored on a type-based level. Specific information, such as the actual position of rings or the ring number, are then stored on occurrence level.

Fig. 6.7

Reuse of recurring elements in the IFC tunnel model (a) using either product representation level (b), representaion level (c) or representation item level (d) [88]

For existing tunnel models, it is also possible to recalculate an optimized model afterwards. Using a rigid body transformation, these recurring geometries of ring segments can be identified and stored in a resource preserving structure.

6.2.5 Interaction Platform

To access all data of a tunneling project in a uniform and standardized way, an interaction platform has been proposed [42]. The interaction platform that considers the TIM framework (Fig. 6.8) consists of three different layers. The first layer contains the actual data sources, such as machine data or borehole profiles. The second layer represents the TIM, containing models that are linked to one or more data sources. Finally, the integration layer provides a unified interface to support a standardized way to use the data for subsequent tunneling software applications. This is realized by constantly using the IFC data format for the transportation of the product models. By using so-called model views, implemented on top of the IFC format using the Model View Definition (MVD) in MVDxml format. This allows to set specific requirements, which must be included into a model to execute a specific application, as well as it allows to filter only the necessary data.

Fig. 6.8

Tunnel information modeling framework based on a unified interaction platform and an application layer (a). The tunnel information model contains four main subdomain models (b). [42]

6.2.6 Visualization of Settlements Based on Analysed Measured Data

An important task for the evaluation of the tunneling process is the analysis of settlements. Based on the reference project ‘‘Wehrhahn-Linie’’ in Düsseldorf, Germany, a case study has been conducted to visualize the amount of settlement during the tunneling process. The settlements must be considered both in the context of the existing built environment and the dynamic position of the current data of the TBM. This allows to identify potentially endangered buildings along the alignment of the tunnel. Also, conclusions about the operational behavior during excavation can be drawn from the settlements, i.e. the values of the excavation advance or grouting pressure can be checked in case of unexpectedly high settlements. For the analysis of settlements, a dedicated model view has been defined (see Fig. 6.9). Only necessary contents are provided, for example, how to provide the built environment, which only is delivered in geometric representation while providing a ground representation is completely discarded in this case. However, the model view ensures, that the required tunnel and the boring machine as well as the settlement values itself are available, before the application can be executed.

Fig. 6.9

Model View Definition for the settlement analysis

Based on the model view, the data was used into a settlement application to analyze and visualize settlements over a period of time corresponding to the excavation process. The visualization is performed step-wise based on a time-step (hours, days and weeks) or ring-wise manner. Based on the current excavation, the installed rings are turned on visible and the position of the machine in front of the tunnel is updated, accordingly. Additionally, the performance data of the boring machine is plotted aside and can be reviewed to the corresponding time step.

The settlement data, which is input in form of 2d-positions and vertical displacement values, is transferred into surface-based representations and colored according to the displacement value. This allows to easily identify spots of endangering settlement values. A surface-based representation, which can be calculated in real-time, is based on Delauney triangulation (Fig. 6.10a). The accuracy of this method is sufficient, when the measurement points are regularly distributed. However, when settlement values occur scattered or spatial varying, this method is not sufficient. In this case, the application of geo-statistical methods help to create a surface representation. In particular, the Simple Kriging method has been applied, which outputs a statistical interpreted value on a regular grid (Fig. 6.10b). This leads to a smoother indication of the settlement values, where relevant spots can easily be identified. Using this method, points must be pre-calculated for a specific moment in time before executing a time-dependent settlement analysis. Therefore, both resource consumption and performance must be considered, as well as data size, which depends on the step size (e.g., number of days or number of rings), the resolution of the generated grid, and the geostatistical interpolation algorithm itself.

Fig. 6.10

Delauney triangulation (a) vs. Kringing method (b)

An example of a specific real-project situation of a time-dependent visualization of settlements is presented in Fig. 6.11. Here, a conspicuous heaving above the TBM could be resolved to a peak in the thrust force value, which originated from the boring machine passing through a bearing slurry wall.

Fig. 6.11

Time-dependent visualization of settlements in context of the existing built environment and the current tunneling advancement

6.3 Digital Design of Tunnel Tracks

The investigation of the tunnel track design is one of the most important studies in tunnel construction. Variations of alignments are compared in order to plan cost-efficiently and to reduce the risks for the project. Usually, variations of alignments are valid, as the criteria allow for alternative approaches that must be evaluated by experts. The Weighted Decision Matrix (WDM) method is used for the evaluation of such variants, by establishing, weighting and comparing project-specific criteria. A manageable number of alignments are examined, until the best alignment is selected for a project. The variants also consider variables in the tunnel design, such as variations in the tunnel diameter, number of tunnel tubes (dual-tube vs. single-tube design) or a strategic selection of methods in tunnel excavation. However, the most important factors are impacts on the environment, such as critical settlements in protected areas or collision with underground structures.

Traditionally, an alignment is composed of a route and a gradient. The route describes the alignment in a 2D plane, while the gradient adds the elevation depending on the current curve section. Both tracks are defined in segments and modeled by special curves section by section. This is necessary, for example, to take into account geometric criteria of elevation and speed. A significant aspect of this approach is the use of transition segments for alignment definition. These are special curve sections, often modeled as clothoid curves, which enable a smooth transition of the curve curvature between segments. The combination of these two design perspectives creates a 3D alignment that can subsequently be used for placement of elements in a BIM-based process.

Fig. 6.12

Integration of interactive planning and evaluation between prediction and risk assessment for the digital alignment planning procedure

A number of tools is required for the investigation of alignment variants. The integrated platform for interactive alignment exploration (Fig. 6.12) has been conceptually developed within the course of the research of the SFB 837. Here, the main focus is on the interactive exploration of alignment variants, new methods for the variant evaluation and the integration of domain-specific information.

6.3.1 Integrated Platform

The Tunnel Information Modeling Framework (TIM) [42] has been adapted into an integrated platform and developed further in [79]. The platform (Fig. 6.13) advances the TIM framework by including alignment planning methods, integrating BIM and GIS data, as well as improving the decision-making by establishing an interactive planning environment for the exploration. The framework has been conceived to utilize roomware products, mainly touch-table devices, to enable a collaborative, interactive and ongoing exploration of planning variants. Therefore, a primary feature of the integrated platform is a (multi-)touch-based interaction with the planning environment that enables updates of planning variants in a collaborative environment in real-time. The creation process of the segmented alignment has been generalized using spline curves, especially Non-Uniform Rational B-Splines (NURBS), in order to create a more flexible representation for alignment definition.

Fig. 6.13

Overview of the Integrated Platform Environment and Processes for Alignment Planning in Tunneling

6.3.2 Interactive Exploration of the Alignment

Alignment planning requires the segmentation and procedural creation of curve sections to create alignments that comply with necessary properties, such as a slowly increasing and decreasing angle of the gradient or a smooth transition of curvature during turns.

Thereby, each segmented curve-type is defined individually, which are primarily a set of lines, circular-arcs, clothoids or parabolas. Handling these curve-types separately, however, complicates the planning process in an interactive environment.

Figure 6.14 displays the step by step approach for alignment modeling. The process starts with a set of planning data, which provides basic constraints to model the alignment on top. The alignment is then refined in each step, by adding more and new types of segments into the alignment definition. In the end, the alignment has to be reviewed against project requirements and constraints, creating a variant for alignment evaluation.

Fig. 6.14

Procedure of planning the route on an alignment [79]

6.3.3 Spline-Based Alignments Planning Approach

Because of their flexibility and consistency, spline curves are often used as a basis for the representation of different types of curves. Whereas lines and circular-arc elements can be displayed quite effortlessly, the implementation of clothoids using splines is much more complex. The main reason for this is that it is difficult to represent the continuously growing curvature of a clothoid curve by splines. This issue has been studied in several articles where different approaches to represent clothoids using splines have been elaborated, e.g., by approximating in a certain interval [90] or parametrically identifying controlled and restrictive computations [89]. This approaches hat been considered for an interactive design of tunnel tracks.

In [79] a set of spline-curves, specifically Non-Uniform Rational B-Splines (NURBS), were investigated to model different types of curves for alignment definition. More specifically, the parameters of these splines had been restricted in specific ways, to emulate the behavior and properties of different curve-types. For example, setting the weights depending on the angle and length of the control polygon can represent circular-arc segments given by start- and end-point. The tangential continuity between segments can then be modeled by splitting the NURBS into subsets of spline-curves or by synchronizing the tangential direction at start- and end-points. While NURBS enable highly adjustable modeling approaches, certain curve-types are modeled as near enough approximations, such as clothoids.

6.3.4 Alignments for Parametric Tunnel-Design

In the context of BIM-based projects, alignments are used for positioning of elements relative to the path and tilt of the alignment axis. For example, in road and rail projects they are used to semi-automatically generate streets- and rails-systems, including technical equipment and information about ablation of ground on the construction site. Considering projects of the tunneling domain, however, alignments are used to generate tunnel models, using a cross-section and design parameters as reference for the modeling process. This process has been evaluated in [78], by using an alignment and cross-section information to generate tunnel models in mechanized tunneling, specifically in segmental ring design (Fig. 6.15). The process of aligning elements to an alignment relies on a sequence of transformations to be applied systematically (see Fig. 6.6).

Fig. 6.15

Generated tunnel model in segmented ring design using an alignment and cross-section information as a basis, applying the method in [78]

These models were exported as Industry Foundation Classes (IFC) models and could be used for information delivery in infrastructure and visualization in the tunnel excavation. Here, prototypical instances of the IFC tunnel extension [15] were used to create semantically correct tunnel models and alignments for data exchange, instead of placeholder elements as is common practice in the industry.

6.3.5 Collaborative Exploration Using Roomware Products

The alignments planning process relies on experts feedback and knowledge. In a collaborative environment, results of the planning design and evaluation can be discussed directly. However, common planning tools and evaluation methods are not designed to provide feedback in real time or to involve a group of stakeholders. Here, the benefits of a collaborative approach can lead to significant improvements in the decision-making process of alignment selection. For example, it can lead to a reduction of the number of planning variants and shorten the development time by directly implementing changes.

To allow a collaborative approach for alignment planning to be applied, the interactive exploration is performed on touch-table devices (see Fig. 6.16, Collaborative Planning Environment). Such a roomware device can be surrounded and interacted with by a group of people. To handle user interactions the planning application running on such a device needs to incorporate (multi-)touch-strategies and streamline the planning process in order to enable semi-automatic reevaluation of the modified alignment. For that purpose a set of multi-touch strategies has been investigated in [79] and implemented into the integrated platform. These touch-strategies were mainly designed to handle simultaneous multi-touch interactions by users (Area-Group) and touch-interactions with elements further away from the users perspective (Lock-On). Therefore, with each interaction the alignment can be modified and planned using only touch, while the updated alignment will be reevaluated in real time and results are visualized for validation (Fig. 6.16, Interactive Planning Tools).

Fig. 6.16

Adopting planning and evaluation tools into a collaborative environment using roomware devices, such as touch-tables and VR-walls, for real time evaluation and live feedback

6.3.6 BIM and GIS Integration

The platform considers the integration of documents and models primarily used in the BIM and GIS domains. These are often considered integrated approaches since their information complements each other in the context of the spatial environment.

The concept of integrating BIM and GIS, however, is often used as an umbrella term that distinguishes between integration approaches in scope and purpose of considered information. For that matter, Beck et al. [6] investigated related research and categorized it based on integration effort and used terminologies. Considering the approaches in [42, 79, 81] the integrated platform falls under the categorization of instance-level information integration of real-world objects that differs in perspective and links based on utilizing queries with spatial reasoning. Also, a multitude of documents and models were investigated for integrated planning. The set of considered documents and models consists of, for example, map data (e.g. cadastrals), built environments (e.g. city model), terrain and ground models (Fig. 6.17). However, the integration of both domains has proven to be challenging due to differences in format, detail, and scope of considered data. Documents and models can vary in level of development and contain a variety of geometric representations.

Fig. 6.17

Documents and model considered in a BIM and GIS integrated planning environment of the tunnel domain

Dealing with geometric differences is a particular challenge since documents and models must be represented superimposed. This requires the handling of transformations of the geo-localization and conversion of the geometry into a uniform representation form in order to be able to perform operations for a comparison. In order to integrate new processes and their representation into the planning process, these must first be aligned with those of the integrated environment. For example, in [47] a voxelized approximation of a ground model has been used for the examination of soil and borehole information. This enabled to quantify uncertainties for the ground model and optimize the placement of new boreholes. Integrating such a model based on existing borehole datasets greatly improves decision making, but may require an alternative representation to comply with the superposition of individual documents and models.

6.3.7 Rule-Based Alignment Evaluation and Data Acquisition

When exploring an integrated environment, rule-based exploration can add value to information acquisition process and for checking constraints and requirements. Considering the superimposed nature of documents and models from BIM and GIS domains allows for spatial reasoning approaches to be applied. In [81] the alignment planning context (see Sect. 6.3.8 for details) also has been incorporated into a reasoning approach, creating context rich relations between elements. This relations were then used to query information across documents and models, which enabled a semi-automated evaluation of relevant requirements and constraints.

Further studies examined the use of a decision model for the choice of tunnel systems (single tube vs. double tube) [86] as well as for the selection of a tunnel boring machine [91].

Sets of elements are highlighted by the queries, which can then be examined in more detail for evaluation. For direct visual feedback, such results can be selected and colorized, which allows for visual validation and subsequent investigations. In such an integrated platform, certain requirements and conditions (e.g. range constraint) can be transformed into geometric representation, which further allows for visual feedback in the investigated area.

6.3.8 Constraints and Requirements

For maintaining a context sensitive approach in tunneling and alignment evaluation, a set of relevant requirements and constraints, that must be met for the creation of valid alignments, are incorporated into the reasoning. These constraints and requirements take into account the correlation between the geologic conditions, the built environment, the parameters of the excavation process, and the operating conditions. Specifically, these can be categorized into the types of

• geometrical/geographical requirements,

• driving dynamics,

• cross section design,

• built environment,

• safety criteria and

• socio-cultural factors.

Geographical and geometrical requirements include criteria such as stations to be connected or existing infrastructure. Requirements of the type of driving dynamics depend mainly on the mode of transport. These include limit values for alignment elements, e.g., maximum radii or the maximum length of straight sections in road tunnels. In addition, the cross-section, a double tube or two single tubes, must be determined. Environmental factors include, for example the protection of FFH areas (Flora-Fauna-Habitat). The last two groups consider e.g., the location of emergency exits for safety criteria and the protection of cultural heritage for socio-cultural factors.

These context-specific requirements are examined and a set of valid variants is formed. These in turn are evaluated using evaluation criteria such as risk assessments and cost estimates (Fig. 6.18). For the evaluation interaction results from settlement analysis (see Sect. 6.2.6) and risk assessment of building damage (see Sect. 6.5.3) can be incorporated.

Utilizing decision models, which consist of rules performing inferences in an axiomatic system of decisions (IF/THEN), can significantly improve the decision-making process for the tunnel domain. In the field of tunnel construction, the application of decision models has been investigated in [83, 84], by investigating the decision-making process of furnishing tunnels with safety systems. This research demonstrated that decision models can enhance the decision-making process and that the weighting of individual criteria influences the evaluation, which also applies to the alignment analysis and selection.

Fig. 6.18

An overview of requirements and constraints affecting the alignment planning process [80]

6.3.9 Utilizing Semantic Web Approaches

Semantic web technologies and approaches can be used to integrate data from the BIM and GIS domains. Using ontology-based data structures, domain-specific information can be reorganized and combined to explore an integrated environment. For that reason, in [81] the ontology for Spatial Reasoning in Tunneling (SRT) has been conceived to enable the investigation of geometries for a spatial reasoning approach. With the SPARQL Protocol and RDF Query Language resulting data structures can be investigated systematically. However, for exploring a spatial environment, considered geometries must be transformed to describe the same spatial context. This includes handling geo-localization of elements and synchronizing their representation in the level of detail and dimension. This is necessary because considering data from the BIM and GIS domains will result in a blend of 2D and 3D representations, which can also be defined in different Coordinate Reference Systems (CRS).

In Semantic Web, geometry is usually represented in Well-Known Text representations (WKT) [37, 65], which is a compact and streamlined version of most significant geometric representations. By utilizing WKT-CRS [64], an extension of WKT-representations to handle geometries defined in different CRS, the geo-localization of such elements can be handled naturally by Semantic Web technologies. The spatial reasoning approach can then be performed by utilized GeoSPARQL [63], an established library and SPARQL extension for investigating spatial relations between geometries. This library uses the Dimensionally Extended 9-Intersection Model method (DE-9IM) [26, 44], which uses a set of distinct intersections to distinguish between geometrical conditions, such as intersect, touch, within, cross, etc.

This approach of utilizing Semantic Web for a linked data and acquisition of relevant information has been validated by numerous publications [35, 8, 87]. Including this approach in the integrated platform enables the semi-automated execution of queries to perform data acquisition and evaluation across documents and models. This concept (Fig. 6.19) has been elaborated and implemented in [81], resulting in the establishment of context-rich relations across documents and models by also including constraints and requirements in the query process.

Fig. 6.19

The method of establishing ontology-based data structures to query across multiple documents and models [81]

6.3.10 Application for Decision-Making and Examination

For decision-making in the alignment selection process, commonly different alignment variants are compared with evaluation criteria using e.g., the Weighted Decision Matrix Method (WDM). The method takes into account a number of constraints and requirements that are evaluated and compared, leading to the selection of a preferred variant, e.g., based on the examination of cost effectiveness [72] (see Sect. 6.3.8). Such an evaluation method is based on the specifics of the planning environment and the associated data. In order to identify relevant data for examination, the spatial environment must be examined. By using Semantic Web technologies, several cases of semi-automated data acquisitions can be performed. In [81], for example, the execution of predefined SPARQL queries enabled the identification of tunneled buildings (Fig. 6.20) or buildings that are under historic preservation and within the vicinity of the alignment (Fig. 6.21). These filtered subsets can be highlighted, providing visual feedback on the results.

Fig. 6.20

Finding and comparing the number of tunneled buildings in single and dual tube design [81, Case 1]

Fig. 6.21

Finding buildings under historic preservation and within the vicinity of the alignment [81, Case 2]

Since these queries accommodate for constraints and requirements of the alignment planning process, they can be incorporated into the decision-making process by integrating results as relevant information for the examination of criteria in a WDM approach. For example, for examining geometric geographical requirements, the identification of buildings within the vicinity of the settlement-affected area is important. Deducing the number and status of those buildings provides a significant resource investigating buildings that are subject to potential damages (see Fig. 6.61). In combination with other factors, such as potential expenses for maintenance and compensation, these query results provide valuable inside for decision-making.

6.4 Process-Oriented Numerical Simulation

This section presents the software that was produced in this project for Advanced Tunneling Engineering (ekate), as well as a Tunnel Analysis Model based on the CutFEM method. Also, a discussion of an automatic model generation based on BIM with aspects of parallelization are given.

6.4.1 ekate: Enhanced KRATOS for Advanced Tunneling Engineering

The simulation model, denoted as ekate (Enhanced Kratos for Advanced Tunneling Engineering), has been implemented via the object-oriented finite element code Kratos [22]. The latter is an open-source framework dedicated to perform numerical simulations for multi-physics problems. Its modular structure provides efficient and robust implementations of various algorithms and schemes (e.g. solution methods, time integration schemes, element formulations, constitutive laws, etc). Kratos is written in C++, in which its kernel provides the basic functionalities and data managements, while, applications characterize the implementation aspects of the numerical model for different physical problems. Herein, the simulation model is developed using Kratos Structural Application and Ekate Auxiliary Application. More detailed discussion about the model can be found in [55], while basic strategies and implementation aspects are presented in [77].

The main goal of the model is to provide an efficient yet realistic simulation environment for all interaction processes occurring during machine driven tunnel construction. Therefore, the model includes all relevant components of the mechanized tunneling process as sub-models, representing the partially or fully saturated ground, the tunnel boring machine, the tunnel lining, hydraulic thrust jacks, the tail void grouting and can take into account soil improvement by means artificial ground freezing, which are interacting with each other via various algorithms. The interaction between the shield and the excavated ground is taken into account via frictional contact algorithm. The shield-lining interaction is described with truss elements (hydraulic jacks) connected between the front surface of the last activated lining segment and the shield, by which, the lining acts as a counter-bearing for the hydraulic jacks thrust to push forward the shield machine. Figure 6.22 shows the basic model components on the left and their respective representation in the finite element mesh on the right. In what follows, the basic model components, the steering algorithm for shield advancement and the simulation script for modeling the construction process are discussed in more detail.

Fig. 6.22

Computational model for mechanized tunneling ekate. left: main components involved in the simulation of the mechanized tunneling process and, right: finite element discretization of the model components; (1) Geological and ground Model, (2) Shield Machine, (3) Tunnel Lining, (4) Tail void grouting and (5) Thrust Jacks [50]

6.4.1.1 Modeling of Ground and Ground Support in Shield Tunneling

Ground model

The ground model is formulated within the framework of the Theory of Porous Media (TPM) [10] that accounts for the coupling between the deformations of the solid phase and the fluid pressures (i.e. an incompressible water phase and a compressible air phase). In that, deformations and pressures are taken as primary variables. The governing balance equations build a set of partial differential equations as the basis of finite element solution. In the following, the two-phase model for fully saturated soils (Fig. 6.23) is briefly presented.

Fig. 6.23

Fully saturated soil modeled according to TPM [50]

The following balance equations prescribe the momentum balance of the mixture, and the mass balance of both solid and fluid phase. Under the assumption of incompressible solid and water phase, the mass balance of each constituent $\alpha$ [$\alpha=s$(olid) or $w$(ater) ] is given by

$$\displaystyle\frac{D_{\alpha}}{Dt}{\rho^{\alpha}}+{\rho^{\alpha}}\text{div}\,\dot{\mathbf{x}}^{\alpha}=0,$$

(6.1)

where $\rho^{\alpha}$ is the average density of a constituent $\alpha$. The porosity $n$, which defines the volume fraction of water, is used to describe the solid and water phases. Therefore, the average density of the mixture $\rho$ can be determined by the intrinsic density of each constituent $\varrho^{\alpha}$ as

$$\displaystyle\rho=\sum{\rho^{\alpha}}=(1-n)\varrho^{s}+n\varrho^{w}.$$

(6.2)

In addition, the velocity of the solid skeleton $(\dot{\mathbf{x}}^{s}=\dot{\mathbf{u}}^{s}=D_{s}\mathbf{u}^{s}/D_{t})$ and the diffusion velocity $(\boldsymbol{\nu}^{ws}=\dot{\mathbf{x}}^{w}-\dot{\mathbf{u}}^{s})$ are used to describe the motion of the constituents. Thus, Eq. 6.1 yields to

$$\displaystyle\frac{D_{s}}{Dt}((1-n)\varrho^{s})+(1-n)\varrho^{s}\text{div}\dot{\mathbf{u}}^{s}=0\qquad\text{for solid skeleton}$$

(6.3)

and

$$\displaystyle\frac{D_{w}}{Dt}(n\varrho^{w})+n\varrho^{w}\text{div}(\boldsymbol{\nu}^{ws}+\dot{\mathbf{u}}^{s})=0\qquad\text{ for pore water}.$$

(6.4)

For Eq. 6.3, assuming incompressible solid grains (i.e. ${D_{s}\varrho^{s}}/{D_{t}}=0$), a differential equation for the porosity can be derived as

$$\displaystyle\begin{aligned}\displaystyle&\displaystyle{-}\varrho^{s}\frac{D_{s}n}{Dt}+(1-n)\varrho^{s}\,\text{div}\,\dot{\mathbf{u}}^{s}=0,\\ \displaystyle&\displaystyle\frac{dn}{dt}=(1-n)\,\text{div}\,\dot{\mathbf{u}}^{s}.\end{aligned}$$

(6.5)

For the mass balance of the water phase, the time derivative with respect to the current configuration of the water phase is transformed to the current configuration of the solid phase as

$$\displaystyle\frac{D_{w}}{D_{t}}(\rho^{w})=\frac{d\rho^{w}}{dt}+\mathop{\mathrm{grad}}\rho^{w}\cdotp\boldsymbol{\nu}^{ws}.$$

(6.6)

The water flow $\tilde{\boldsymbol{\nu}}^{ws}$ through the pore spaces is described by Darcy’s law [23]. Accordingly, the flow is governed by the pressure gradient and the volume of pore spaces and expressed as

$$\displaystyle\tilde{\boldsymbol{\nu}}^{ws}=-\dfrac{k^{w}}{\varrho^{w}g}(\mathop{\mathrm{grad}}P^{w}-\varrho^{w}\mathbf{g}),$$

(6.7)

where $k^{w}$ is the hydraulic conductivity that symbolizes the available pore spaces in soil. Using the volume fraction $n$, the Darcy’s velocity is related to the diffusion velocity $\boldsymbol{\nu}^{ws}$ as $\tilde{\boldsymbol{\nu}}^{ws}=n\,\boldsymbol{\nu}^{ws}$. Applying Eq. 6.6 to Eq. 6.4, the mass balance of the incompressible water phase can be written as

$$\displaystyle\begin{aligned}\displaystyle\frac{d}{dt}(n\varrho^{w})+\mathop{\mathrm{grad}}(n\varrho^{w})\cdotp\boldsymbol{\nu}^{ws}+n\varrho^{w}\text{div}(\boldsymbol{\nu}^{ws}+\dot{\mathbf{u}}^{s})=0,\\ \displaystyle\frac{d}{dt}(n)+\text{div}(\tilde{\boldsymbol{\nu}}^{ws})+n\,\text{div}(\dot{\mathbf{u}}^{s})=0,\\ \displaystyle\text{div}(\tilde{\boldsymbol{\nu}}^{ws})+\text{div}(\dot{\mathbf{u}}^{s})=0.\end{aligned}$$

(6.8)

The second balance relation is introduced by the overall momentum balance of the mixture using the averaged Cauchy stress $\boldsymbol{\sigma}$ as

$$\displaystyle\text{div}(\boldsymbol{\sigma})+\rho\mathbf{g}=\mathbf{0},$$

(6.9)

According to [82], the effective stresses define the inner grain interaction (i.e. the stress-strain behavior of the soil skeleton). The effective stresses in a fully saturated soil are determined as

$$\displaystyle\boldsymbol{\sigma}^{s}{{}^{\prime}}=\boldsymbol{\sigma}+P^{w}\mathbf{I},$$

(6.10)

where the effective stresses $\boldsymbol{\sigma}^{s}{{}^{\prime}}$ and the water pressure $P^{w}$ are the stress variables and $\mathbf{I}$ denotes the unity tensor.

The mass balance and the momentum balance equations form the set of partial differential equations to be solved in which the deformations and water pressures are the primary field variables. Further discussion regarding the multi-phase model for partially saturated soils and its numerical implementation has been presented in [55].

The material behavior of the soil skeleton is represented by means of nonlinear elasto-plastic constitutive laws; namely Drucker-Prager (DP) law or Clay And Sand Model (CASM) [93]. DP-law presents a relatively simple model based on the approximation of the Mohr-Coulomb criteria using a smooth yield function, see Fig. 6.24, left. A generalized behavior for both clay and sand soils can be modelled by CAS-model. Figure 6.24, right, shows the yield surface in the principal stress space. The latter is similar to the Cam-Clay models, yet, it overcomes the limitation of Cam-Clay models for the characterization of sands and highly over-consolidated clays.

Fig. 6.24

Yield function in principal stress space and in the $p^{\prime}$-$q$ plane: Drucker-Prager-model (left) and Clay And Sand-model (right) [50]

Grouting mortar

The annular gap between the tunnel lining and the excavated ground is filled simultaneously with a pressurized grouting mortar, Fig. 6.25a. The latter is a mixture that consists of a hyper-concentrated two phase material [9]. It should maintain, at the early stage, a certain degree of workability to be distributed uniformly around the lining. On the other hand, hardening should occur to resist the buoyancy of the lining and to prevent the dislocation of the joints. The setting of grouting mortar is characterized by an increase of mechanical stiffness accompanied with a phase change from semi-liquid to solid state, Fig 6.25b.

Fig. 6.25

Annular gap grouting. a sketch of annular gap grouting through a nozzle in shield skin and b the process of grouting mortar hydration with stiffness and permeability evolution [50]

To model the pressurized grouting mortar, a two-phase (hydro-mechanical) formulation is used, which is similar to the finite element formulation of the ground model. The grouting pressure is applied as pore water pressure to the fresh mortar. Stiffening of the grouting mortar is considered by a time-dependent hyper-elastic material and time-dependent permeability to account for the hydration process. Simultaneous grouting of the annular gap is simulated by the step-wise activation of the corresponding grouting mortar elements with respect to current shield position, while pressurization is realized by a prescribed pressure boundary condition on the face of the elements at the shield tail.

Herein, an exponential relation is used to define the temporal evolution of permeability. This assumption has been already proposed in [41]. The permeability of the grouting element is updated at the beginning of each time step, where the mathematical expression is given by

$$\displaystyle k(t)=(k^{(0)}-k^{(28)})e^{-\beta_{\text{grout}}t}+k^{(28)},$$

(6.11)

where $k^{(0)}$ and $k^{(28)}$ are the initial permeability and final permeability after 28 days, $t$ expresses the age in hours and $\beta_{\text{grout}}$ is a parameter that controls the change with respect to time. Figure 6.26a shows the time dependent permeability for two different analysis parameters ($\beta_{\text{grout}}=0.05$ and 0.10). With respect to the stiffness evolution of such cementitious material, the proposed material model follows the basic methodology of hyperelasticity for aging materials, as presented in [52, 53], see Fig. 6.26b.

Fig. 6.26

Development of grouting mortar properties with time. a permeability evolution for two different analysis parameters and b description of the parametric function $\beta_{E}(t)$ where the grout is fully hardened after 28 days [50]

For the time-dependent increase of elastic modulus, an irrecoverable strain necessarily occurs. Therefore, the strain tensor $\varepsilon$,

$$\displaystyle\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}^{e}+\boldsymbol{\varepsilon}^{t},$$

(6.12)

is decomposed into a recoverable elastic part $\boldsymbol{\varepsilon}^{e}$ and a non-recoverable part $\boldsymbol{\varepsilon}^{t}$ associated with the time-dependent hydration.

According to the theory of hyperelasticity, a time-dependent function of the stored energy defines the stiffening effect and consequently the time-dependent stress tensor as

$$\begin{aligned}W(\boldsymbol{\varepsilon},t) & =\frac{1}{2}(\boldsymbol{\varepsilon}-\boldsymbol{\varepsilon}^{t}):\mathsf{C}^{28}:(\boldsymbol{\varepsilon}-\boldsymbol{\varepsilon}^{t}),\end{aligned}$$

(6.13)

$$\begin{aligned}\boldsymbol{\sigma} & =\dfrac{\delta W}{\delta\boldsymbol{\varepsilon}^{e}}=\mathsf{C}^{28}:(\boldsymbol{\varepsilon}-\boldsymbol{\varepsilon}^{t}),\end{aligned}$$

(6.14)

where $\mathsf{C}^{28}$ is the standard elasticity tensor of the hardened material in which the superscript $(28)$ indicates a reference time in days at the end of the aging process. The time-dependent material tensor $\mathsf{C}(t)$ is expressed by the development of the Young’s Modulus $E(t)$ as

$$\displaystyle\mathsf{C}(t)=\mathsf{C}^{28}\frac{E(t)}{E^{(28)}},$$

(6.15)

and the experimental observations shows that the stress rate is related to the strain rate by the time-dependent material tensor,

$$\displaystyle\dot{\boldsymbol{\sigma}}=\mathsf{C}(t):\dot{\boldsymbol{\varepsilon}}.$$

(6.16)

The stress increment $\Updelta\boldsymbol{\sigma}$ for a certain time interval $[t_{n},t_{n+1}]$ is determined from the time integration of Eq. 6.16

$$\displaystyle\Updelta\boldsymbol{\sigma}=\int^{t_{n+1}}_{t_{n}}\mathsf{C}(t):\dot{\boldsymbol{\varepsilon}}\,dt=\dfrac{1}{E^{(28)}}\mathsf{C}^{28}:\dfrac{\Updelta\boldsymbol{\varepsilon}}{\Updelta t}\int^{t_{n+1}}_{t_{n}}E(t)\,dt=\dfrac{\chi}{E^{(28)}\Updelta t}\mathsf{C}^{28}:\Updelta\boldsymbol{\varepsilon},$$

(6.17)

in which, $\chi$ expresses the integration of time-dependent Young’s Modulus over the time interval $[t_{n},t_{n+1}]$. Comparing Eq. 6.17 with the incremental form of Eq. 6.14, the incremental time-dependent strain yields

$$\displaystyle\Updelta\boldsymbol{\varepsilon}^{t}=\bigg(1-\dfrac{\chi}{E^{(28)}\Updelta t}\bigg)\Updelta\boldsymbol{\varepsilon}.$$

(6.18)

The elastic algorithmic tangent $\mathsf{A}^{el}$,

$$\displaystyle\mathsf{A}^{el}=\dfrac{\partial\boldsymbol{\sigma}_{n+1}}{\partial\varepsilon_{n+1}}=\dfrac{\chi}{E^{(28)}\Updelta t}\mathsf{C}^{28},$$

(6.19)

can be obtained by the linearization of Eq. 6.14 after inserting Eq. 6.18.

The time-dependent stress-strain behavior of the proposed material is mainly related to the time-variant Young’s modulus. The later is expressed as $E(t)=\beta_{E}(t)E^{(28)}$, see Fig. 6.26b. The coefficient $\beta_{E}(t)$ is defined, according to [53]

$$\displaystyle\beta_{E}(t)=\left\{\begin{aligned}\displaystyle\beta_{E}^{\mathrm{I}}&\displaystyle=c_{E}t+d_{E}t^{2}&\displaystyle&\displaystyle\text{for }t\leq t_{E}\\ \displaystyle\beta_{E}^{\mathrm{II}}&\displaystyle=\Big(a_{E}+\frac{{b_{E}}}{t-\Updelta t_{E}}\Big)^{-0.5}&\displaystyle&\displaystyle\text{for }t_{E}<t\leq 672h\\ \displaystyle\beta_{E}^{\mathrm{III}}&\displaystyle=1.0&\displaystyle&\displaystyle\text{for }672h<t_{E}\end{aligned}\right.$$

(6.20)

where $a_{E},\,b_{E},\,c_{E}\,\text{and}\,d_{E}$ are material dependent parameters determined by the ratio $E^{(1)}/E^{(28)}$, the initial time interval $t_{E}$ and the time step $\Updelta t_{E}$, see [53] for more details.

Face support pressures

Numerically, two scenarios can be characterized by a membrane model and a penetration model. They are described in the model by applying adequate boundary conditions, see Fig. 6.27. For the so called membrane model, where a perfect filter cake is formed, the fluid flow is set to zero and a prescribed total pressure is applied at the tunnel face as

$$\displaystyle\mathbf{t}=p^{\text{sup}}\,\mathbf{n}\leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\ q^{w}=0.$$

(6.21)

For the penetration model, i.e. without a filter cake, both fluid pressure and total stresses are prescribed at the tunnel,

$$\displaystyle\mathbf{t}=p^{\text{sup}}\,\mathbf{n}\leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\ p^{w}=0.$$

(6.22)

The numerical description of the grouting pressure is achieved in a way similar to the description of face support. At the last face of the newly activated elements, the total stresses and water pressures are prescribed as a linear function using the average grout pressure at the tunnel axis $\bar{p}^{\text{grouting}}_{\text{ax}}$ and its gradient $\mathop{\mathrm{grad}}\bar{p}^{\text{grouting}}$,

$$\displaystyle\bar{p}^{\text{grouting}}=\bar{p}^{\text{grouting}}_{\text{ax}}+z\mathop{\mathrm{grad}}\bar{p}^{\text{grouting}}\,,$$

(6.23)

where $z$ is the distance in the direction of gravity, measured from the tunnel axis. Consequently, the total pressure and the water pressure at the element face can be defined as

$$\displaystyle\begin{aligned}\displaystyle p^{w}&\displaystyle=\bar{p}^{\text{grouting}}\,,\\ \displaystyle\mathbf{t}^{\star}&\displaystyle=-\bar{p}^{w}\mathbf{n}\,,\end{aligned}$$

(6.24)

where $\mathbf{n}$ is the normal vector to the grouting face. With the previous description, the effective stresses at the last grouting face, where grout injection is executed, are set to zero.

Fig. 6.27

Prescribed boundary conditions of face support pressure. a Stresses within the two phase element (total stresses $\sigma$, effective stresses $\sigma^{s}{{}^{\prime}}$, partial solid stresses $\sigma^{s}$ and water pressure $p^{w}$), b formation of an impermeable filter cake (Eq. 6.21) and c penetration model with no filter cake sealing the tunnel face (Eq. 6.22) [50]

6.4.1.2 Segment-Wise Lining Installation in ekate

Shield tunnel linings are constructed by an assembly of segments into a complete ring. The simulation of the segmental lining model including the joints is similar to the continuous model to some extent. The main difference is the assignment of the contact interactions at the joints between the segments. The generation of the segmental lining model starts with the consideration of a single ring as shown in Fig. 6.28. The ring model, generated by GiD, consists of volume elements which represent different segments in which the boundary surfaces of each segment are separately defined. Figure 6.28 shows a ring that consists of 7 segments with equal size including bolts and dowels on joints. Each longitudinal joint contains two bolts at the center line, while each segment has two shear dowels in the ring joint. In total, 14 bolts and 14 dowels are used in each ring. The exact joint geometry is not described in this model since the global structural response is the main interest of this study. As such, the effect of the rubber sealing gasket is not explicitly considered. In addition, only joints with flat contact surfaces will be addressed.

Fig. 6.28

ekate representative model for segmental lining geometry including bolts and dowels [50]

The contact interactions between the segments in one ring and between consecutive rings are shown in Fig. 6.29. The contact algorithm is used to characterize the response of joints; this requires the definition of master and slave surfaces for the possible contact surfaces. Since, the complete lining model consists of a large number of joints, there will consequently exist a large number of master and slave contact surfaces. Therefore, each pair of contacting surfaces is associated with a distinct contact index, as indicated in Fig. 6.29, to speed up the search algorithm.

Fig. 6.29

Definition of contact surfaces between the segments as defined in the numerical model [50]

Beam elements with elastic material properties are used to represent the dowels and bolts in the joints. The geometrical properties of the beam are defined according to the corresponding diameter, length and type, as shown in Fig. 6.30. Pre-stressing in the bolts can be considered by applying a certain pre-stressing force for the corresponding element. In finite element formulation, the internal force vector for the beam element with pre-stressing is defined as

$$\displaystyle\mathbf{R}^{\text{int}}=\mathbf{K}\cdot\mathbf{u}+\mathbf{F}^{\text{Prestress}}.$$

(6.25)

The assigned properties to the beam elements describe the desired structural behavior of the elements. For the simulation of shear dowels, only the shear stiffness is required while the axial stiffness can be omitted by setting $A_{\text{axial}}=0.0$. Bolts are mainly simulated by considering axial stiffness and the pre-stressing force if required, while the flexural stiffness can be either considered or ignored. Generally, the dowels/bolts are assumed to act at the center line of the joints and therefore it is not expected that they contribute significantly to the overall flexural stiffness of the lining ring.

Fig. 6.30

Representation of bolts and dowels in segmental lining joints [50]

The physical interaction between the segments and the dowels/bolts is accounted for by using the node to volume tying, see Fig. 6.30. The nodes at the ends of each bolt are embedded in their corresponding volume elements. In the finite element code Kratos, the Embedded″​″​Point″​″​Lagrange″​″​Tying″​″​Utility is used for setting these tying constraints. Within the simulation script, a function, Initialize″​″​Embedded″​″​Point″​″​Lagrange″​″​Tying, sets the tying condition between the end node of the beam ($X^{i}$) and the volume element containing that point. First, the local coordinates $\xi(X^{i})$ at the point location inside the volume element are determined. Then, the condition ties the displacements between the point and its projection inside the volume elements using the Lagrange multiplier,

$$\displaystyle(\mathbf{u}^{\text{node}}-\mathbf{u}^{\text{vol}}(\xi(X^{i})))\cdot\boldsymbol{\lambda}=\mathbf{0}\,,$$

(6.26)

and the tying condition is added to the system of equations,

$$\displaystyle\begin{bmatrix}\mathbf{K}^{\text{node}}&\mathbf{0}&\mathbf{1}\\ \mathbf{0}&\mathbf{K}^{\text{node}}&\mathbf{-1}\\ \mathbf{1}&\mathbf{-1}&\mathbf{0}\end{bmatrix}\cdot\begin{bmatrix}\mathbf{u}^{\text{node}}\\ \mathbf{u}^{\text{vol}}\\ \boldsymbol{\lambda}\end{bmatrix}=\begin{bmatrix}\mathbf{R}^{\text{node}}\\ \mathbf{R}^{\text{vol}}\\ \mathbf{0}\end{bmatrix}.$$

(6.27)

The full description of the numerical model requires the definition of other properties such as material, activation levels, etc., as well. A python script is created to automatize the model generation. First, it imports the geometry of the user defined segmental ring. Then, the segmental ring is placed in its location and rotated according to the required staggered joint pattern. The properties and boundary conditions associated with the lining ring are assigned. The desired joint pattern can be generated independently of the finite element discretization of the ground. Figure 6.31 shows a staggered and aligned configurations of lining joints.

Fig. 6.31

Different joint arrangements in segmental tunnel lining model [50]

It should be noted that the staggered configurations of longitudinal joints are usually preferred in common tunneling practice. In [29], it is explicitly stated that an offset, by half or third of the segment length, should be considered to prevent continuous longitudinal joints across multiple rings. This in return strengthens the lining stiffness and the sealing effect. Moreover, it is not suggested to have the hydraulic jacks pads at the location of longitudinal joints. Therefore, the proposed position of joints is adopted in such a manner that they do not match the position of hydraulic jacks.

Lining-soil interaction

The relation between the outer boundary of the segment and the surrounding grouting mortar requires a particular consideration in the case of explicit modeling of the segmental lining. As shown in Fig. 6.32, the assumption of mesh compatibility with nodal connectivity between the lining and the grouting is only valid for a continuous lining model. To enable segment-wise ring installation, and the correct kinematics of the joints, the connection between the lining outer surface and the grouting material is modeled by means of a surface-to-surface tying procedure, which does not not require mesh compatibility. The tying constraint preventing the relative displacements is enforced at the Gauss points using a penalty approach. The energy functional associated with the penalty term is defined as

$$\displaystyle\Pi^{\text{Tying}}(\mathbf{u})=\dfrac{1}{2}\,\epsilon\,\|(\mathbf{u}^{\text{lining}}-\mathbf{u}^{\text{grout}})\|^{2},$$

(6.28)

where $\epsilon$ denotes the penalty parameter.

Fig. 6.32

Modeling of lining-soil interactions for the continuous (left) and the segmental lining (right) [50]

6.4.1.3 TBM Steering and TBM-Soil Interaction

The shield is modeled as an independent, deformable body that interacts with the excavated soil along its outer surface by means of frictional contact conditions. The shield geometry is depicted according to its design, see Fig. 6.33. The respective FE model as shown in Fig. 6.34 accounts for the main structural and load carrying components (i.e. the shield skin, the shield wall and other stiffening parts). Shield weight including the machinery parts are accounted for. The load is distributed on shield front, along approximately two third of the total length, considering the fact that most of the heavy parts are located at the front. The cutting wheel is not modeled explicitly, instead, the equivalent cutting forces in addition to the face pressure are applied on the shield wall. In addition, overcutting and shield skin tapering are explicitly considered, see Fig. 6.34. This is beneficial for a reliable prognoses of the ground settlements, as well, the adequate prediction of the shield soil interaction is feasible, in particular for curved alignments.

Fig. 6.33

Illustration of the main aspects related to the numerical representation of the shield machine: main structural components represented by the thick black lines (left) and radial distribution of hydraulic jacks (right) [50]

Fig. 6.34

Finite element mesh of the shield machine, the hydraulic jacks and the lining, and the geometrical parameters involved in the definition of the shield model [50]

The hydraulic jacks are represented by Crisfield truss elements [21], that produce the mutual interaction between the shield and the lining and by which shield advancement is achieved. In this context, a steering algorithm is developed to fully automatize the shield movement [2]. The steering algorithm controls the elongation of each hydraulic jack. Prescribed strains and the counter-bearing produced by tunnel lining provides the momentum to move the shield forward. In addition, the steering algorithm includes a correction vector that allows for counter-steering against weight-induced dropping of the shield and keeps the path of the shield on the prescribed tunnel alignment.

The frictional contact characterizes the interaction between the shield skin and the excavated ground. Following the basic concepts in [45, 76], Kuhn-Tucker condition is applied, which defines the separation or direct contact between surfaces. As a result, the simulation model can predict the contact condition between the shield skin and the ground (i.e. whether a gap exists or not). Slave and master contact faces are assigned to shield skin and excavation boundaries respectively, where the Augmented Lagrange method enforces the contact constraint.

In the simulation model, the governing equations are the weak form of the mass balance equation for the ground water flow and the weak form of the equilibrium equation. Since large movements are required for the positioning of the shield, total Lagrangian FE formulation is used for shield discretization. It should be highlighted that the inertial forces are neglected since the machine advances through the soil with low speed. The final position and orientation of the shield results from the force balance on the shield, where the advancement process is achieved by the elongations of the hydraulic jacks.

6.4.1.4 Computational Modeling for Mechanized Tunneling Process

In tunneling simulations, the size of the domain should be chosen in a way that the model boundaries do not affect the results in the tunneling vicinity. Generally, the primary state of stresses at the boundaries should not change [68]. Figure 6.35 shows the ground domain with the prescribed boundary conditions for the simulation of a fully saturated soil using two phase formulation. These boundary conditions remain unchanged during the simulation.

Fig. 6.35

FE mesh of the ground with boundary conditions for the displacement components $u_{x}$, $u_{y}$, $u_{z}$ and pore pressure $P_{w}$ [50]

To account for the primary stress state in the soil, the respective values can be either explicitly given to the model or implicitly determined. In the simulation model, the second approach is adopted, in which a two-steps procedure is followed in the beginning of the analysis. In the first step, the ground model is analyzed under its self weight with the aforementioned boundary conditions. At this point, the ground is assumed to behave elastically and all the other model components are deactivated. The output stresses of this step correspond to

$$\displaystyle\begin{aligned}\displaystyle\sigma{{}^{\prime}}_{z}&\displaystyle=\sigma_{z}+u_{w};&\displaystyle&\displaystyle\text{where}&\displaystyle\sigma_{z}&\displaystyle=-\gamma_{\text{sat}}h\quad\text{and}\quad u_{w}=\gamma_{w}h_{w},\\ \displaystyle\sigma{{}^{\prime}}_{x}&\displaystyle=\sigma{{}^{\prime}}_{y}=K_{0}\sigma{{}^{\prime}}_{z};&\displaystyle&\displaystyle\text{where}&\displaystyle K_{0}&\displaystyle=\nu/(1-\nu).\end{aligned}$$

(6.29)

Using the InsituStressUtility, the stresses at the Gauss points are transmitted as pre-stresses. In this utility, a predefined value for $K_{0}$ can be imposed. Then, the second step solves the equilibrium equation with gravitational loading and pre-stressing. The output ground deformation is checked to ensure that it yields to zero, while the in-situ state of stress is preserved inside the ground.

The aforementioned scheme serves as a basis to determine the primary stress state, that is followed by preliminary steps as shown in Fig. 6.36. These steps start with the initialization of the contact analysis. The shield is activated and positioned at its starting location and the excavated ground is deactivated (Fig. 6.36a). In addition, the face pressure and grouting pressures are applied. Then, the hydraulic jacks are initialized and the shield is allowed to deform. The face pressure and cutting forces are applied on the shield. That leads to evaluation of shield deformation taking into account the contact forces from the ground, the applied loads and its self weight (Fig. 6.36b).

Fig. 6.36

Preliminary steps at the beginning of the simulation of mechanized tunneling. a Initial position of the shield at the model boundary with the initialization of contact analysis, b shield with free deformation supported by the soil pressure and the hydraulic jacks, situation before the start of step-wise simulation [50]

Eventually, the step-by-step simulation is carried out as shown in Fig. 6.37. This is achieved by the repetition of two simulation steps: an excavation step and a ring construction step following the predefined time step for each. The excavation step includes the use of the SteeringUtility to position the shield. This movement is accompanied with the deactivation of the soil and the activation of the grout. Afterwards, the ring construction step is performed by the activation of the lining ring inside the shield accompanied with the resetting of the hydraulic jacks elements on the face of the newly installed ring. Once the shield reaches the final excavation step, the simulation stops.

Fig. 6.37

Repetitive scheme for the step-wise simulation of mechanized tunneling process. a Stand still position, b shield advancement and soil excavation achieved by means of the steering algorithm and the de/re-activation of the respective elements, c ring construction and resetting of the hydraulic jacks [50]

6.4.1.5 Computational Modeling of Artificial Ground Freezing in Tunneling

Artificial ground freezing (AGF) is a ground improvement technique which is used to stabilize the soil to provide temporary support and water flow control. In urban environments with settlement-sensitive buildings, AGF is used to provided temporary ground support during tunnel construction. In order to simulate the ground improvement in tunneling by means of AGF, a computational model was developed for the numerical simulation of coupled thermo-hydro-mechanical behavior of soil upon freezing [95]. The freezing soil computational model adopts the theory of poromechanics [20] where the solid particles, liquid water and crystal ice are considered as three separated phases in conjunction with the theory of premelting dynamics.

Eulerian liquid and ice saturations

At all times the porous volume is assumed to be filled by water, in both liquid form (L) and crystal form (C). Hence, the current Lagrangian porosity $\phi$ can be written as

$$\displaystyle\phi=\phi_{\text{L}}+\phi_{\text{C}}\,,$$

(6.30)

where $\phi_{J}$ is the current Lagrangian partial porosity related to phase $J=\text{L, C}$. Once the overall porosity is known, the current partial porosities can be expressed in terms of the degree of saturation as

$$\displaystyle\phi_{J}=\phi\,\chi_{J}\,,\quad\text{with}\quad\chi_{\text{L}}+\chi_{\text{C}}=1\,,$$

(6.31)

where $\chi_{J}$ denotes the Eulerian saturation and represents the current partial saturation of phase $J$ relative to the current deformed porous volume $\phi\mathrm{D}\Omega_{0}$, see Fig. 6.38.

Fig. 6.38

Schematic illustration of three-phase freezing soil with averaging principle applied [95]

The liquid saturation curve

By analogy with a liquid-gas interface for unsaturated soils and adopting the van Genuchten capillary curve, a relationship between the liquid saturation in freezing soils and temperature can be obtained,

$$\displaystyle\chi_{\text{L}}=\left(1+\left(\frac{T_{\text{f}}-T}{\Updelta T_{\text{ch}}}\right)^{\frac{1}{1-m}}\right)^{-m}\,,$$

(6.32)

where $\Updelta T_{\text{ch}}=\frac{\mathcal{N}\,\gamma_{\text{CL}}}{S_{\text{f}}\,\gamma_{\text{GL}}}$ is the characteristic cooling temperature related to the most frequently encountered pore radius $R_{\text{ch}}$, $\mathcal{N}$ the capillary modulus, $\gamma_{\text{CL}}$ the liquid-crystal interface energy, $\gamma_{\text{GL}}$ the liquid-air interface energy, $S_{\text{f}}$ the freezing entropy per unit of volume and $m$ is an index indicating the pore radius distribution around $R_{\text{ch}}$. The influence of $\Updelta T_{\text{ch}}$ and $m$ on the shape of the liquid saturation curve is illustrated in Fig. 6.39.

Fig. 6.39

Liquid saturation curve during freezing: Influence of $\Updelta T_{\text{ch}}$ (left) and $m$ (right), [95], [94]

The liquid-crystal equilibrium relation

Thermodynamic equilibrium between the liquid pore water (L) and the adjacent crystal ice (C) requires the equality of the chemical potential of both phases,

$$\displaystyle\text{d}\mu_{L}=\text{d}\mu_{C},\ \text{with}\ \text{d}\mu_{J}=\frac{\text{d}p_{J}}{\rho_{J}}-s_{J}\text{d}T\,\quad(J=\text{L,C}),$$

(6.33)

where $\mu_{J}$, $p_{J}$, $\rho_{J}$ and $s_{J}$ are the chemical potential, the pressure, the specific mass density and the entropy per unit mass of phase $J$, respectively. Integrating the above equation between the reference state $p=$ 0 Pa, $T=T_{\text{f}}=$ 273 K and the current state provides the liquid-crystal equilibrium, we have

$$\displaystyle p_{\text{C}}-p_{\text{L}}=S_{\text{f}}\left(T_{\text{f}}-T\right)\,,\text{with}\ S_{\text{f}}=\frac{\rho_{c}L_{\text{f}}}{T_{\text{f}}}.$$

(6.34)

Note that Eq. 6.34 can be used to explain the micro-cryo-suction mechanism, which is identified as the driving force of frost heave phenomenon observed for frost-susceptible soils.

The three-phase finite element model for freezing soils

The computational model is a thermo-hydro-mechanical three-phase finite element model which considers the temperature, liquid pressure and solid displacements as the primary variables. The three-phase finite element model captures the most relevant couplings between the phase transition associated with latent heat effect, the liquid transport within the pores, and the accompanying mechanical deformation trough three fundamental physical laws, which are the overall entropy balance, mass balance of liquid water and crystal ice, and overall momentum balance, together with corresponding state relations.

Mass balance of liquid water and crystal ice

Considering the possible phase transition between liquid water and crystal ice, the mass balance equation relative to each phase can be written as

$$\displaystyle\frac{dm_{\text{L}}}{dt}+\nabla\cdot\mathbf{w_{\text{L}}}=-\overset{\circ}{m}_{\text{L}\to\text{C}}\,,\quad\frac{dm_{\text{C}}}{dt}+\nabla\cdot\mathbf{w_{\text{C}}}=\overset{\circ}{m}_{\text{L}\to\text{C}}\,,$$

(6.35)

where $m_{J}=\rho_{J}\,\phi_{J}$ represents the current mass content related to phase $J$ per unit of initial volume, with $\rho_{J}$ being the corresponding specific mass density; $\nabla\cdot(\,{\cdot}\,)$ denotes the divergence operator; $\mathbf{w}_{J}$ is the Eulerian relative mass flow vector, and $\overset{\circ}{m}_{\text{L}\to\text{C}}$ is the rate of liquid water mass changing into crystal ice. With the assumption that the flow of ice with respect to the skeleton occurs much slower than the flow of water such that $\mathbf{w}_{\text{C}}=\mathbf{0}$, summation of the mass balance equations yields the combined mass balance for both the liquid water and the crystal ice phase. The mass balance equation for the ice and liquid phases reads

$$\displaystyle\frac{dm_{L}}{dt}+\frac{dm_{C}}{dt}+\nabla\cdot\mathbf{w}_{L}=0.$$

(6.36)

Overall momentum balance

Neglecting dynamic effects, the momentum balance equation for the mixture is given as

$$\displaystyle\nabla\cdot\boldsymbol{\sigma}+\rho\,\mathbf{g}=0,$$

(6.37)

where $\boldsymbol{\sigma}$ denotes the tensor of total stresses, $\rho=(1-\phi_{0})\,\rho_{\text{S0}}+m_{\text{L}}+m_{\text{C}}$ stands for the overall mass density with $\rho_{\text{S0}}$ being the initial mass density of solid particles, and $\mathbf{g}$ is the gravity force per unit volume.

Overall entropy balance

Identifying the spontaneous production of entropy $\Phi_{M}$, the second law of thermodynamics states the entropy balance for the liquid-ice crystal-solid mixture:

$$\displaystyle T\left(\frac{dS}{dt}+\nabla\cdot(s_{L}w_{L})\right)+\nabla\cdot\mathbf{q}-\Phi_{M}=0,$$

(6.38)

with $S=S_{\text{S}}+m_{\text{L}}\,s_{\text{L}}+m_{\text{C}}\,s_{\text{C}}$ as the overall density of entropy per unit of volume, while $S_{\text{S}}$ is the entropy of the solid matrix and $s_{J}$ the specific entropy related to phase $J$, $\mathbf{q}$ is the overall outgoing heat flow vector. Here $\Phi_{\text{M}}$ represents only the mechanical dissipation associated with the viscous liquid flow through the porous volume.

Numerical simulation of AGF under seepage flow

The computational modeling of artificial ground freezing (AGF) method represents a challenge due the hydro-thermo-mechanical interactions between the frozen and unfrozen surrounding soil. In tunneling application, the developed computational was used to investigate the influence of horizontal seepage flow on the formation of a frozen arch wall during AGF and an optimization of freeze pipe arrangement was investigated with the goal of speeding up the time to obtain a fully frozen arch, see [51]. Figure 6.40 compares the spatial distribution obtained from three different levels of seepage flow $\mathbf{v}_{L}$ = 0, 0.5 and 1.0 m/d after 3, 6 and 9 days of continuous freezing.

Fig. 6.40

Influence of seepage flow on the formation of a frozen arch, [51], [50]

Application of ant colony optimization to find an optimized arrangement of the freeze pipes

The results of Fig. 6.40 lead to the conclusion that a large seepage flow will significantly delay or even avoid the formation of a closed frozen arch around the tunnel profile during freezing process. In such situations, the frost zone around the freeze pipe does not form concentrically around the freeze pipes. It is clear, that an equidistant distribution of the freeze pipes is not the optimal arrangement in case of presence of seepage flow. The success of the freezing process may be endangered in a situation when a steady state is reached without forming a closed frozen arch. For this reason, the groundwater flow must be adequately considered in the design of the freezing operation to achieve a successful freezing process.

In order to enhance the freezing efficiency, the Ant colony optimization (ACO) described herein has been applied to search for the optimal arrangement of freeze pipes depending on the direction and magnitude of the groundwater flow. ACO algorithm is a probabilistic method with the goal to search the optimal path in a graph by mimicking the behavior of ants seeking a path between their colony and a source of food, see [30]. The artificial ant is a simple computational agent of the ACO iterative algorithm. In Fig. 6.41 a walking ant on the graph simulates the solution selection process. At each iteration of the ACO algorithm, each ant moves from a solution state to another solution state creating a partial solution until it constructs the complete solution.

Fig. 6.41

Ant Colony Optimization: Illustration of discrete solution space, of note is that each column resembles one variable including the set of attached discrete values, [51], [50]

The results after optimization show that an arrangement of the freeze pipes determined by the optimization algorithm considerably reduces the freezing time for the formation of a frozen arch. With increasing seepage velocity, the freezing time increases progressively in case of an even distribution of the freeze pipes. In contrast, only a moderate increase is observed if an optimal placement is chosen. It is notable that the larger the flow velocity of the groundwater, the larger is the improvement obtained from the optimization procedure. The optimum arrangement of freeze pipes is presented in Fig. 6.42 for $\mathbf{v}_{\text{L}}=1.0$ m/d. In Fig. 6.42, note that the pipe locations are shifted against the seepage flow direction and that the spacing between pipes is decreased at the upstream direction. For a seepage velocity of $\mathbf{v_{\text{L}}}=1.0$ m/d, the optimized solution requires a freezing time of $10$ days to form a fully frozen arch. In contrast, to the original design with an equidistant placement of the freeze pipes, for which more than $50$ days of freezing are required. Figure 6.43 shows the formation of the frozen arch by means of the temperature distribution at different days of the freezing process for an optimized placement of the pipes for a seepage velocity of $\mathbf{v}_{\text{L}}=0.5$ m/d and $\mathbf{v}_{\text{L}}=1.0$ m/d, respectively. When the optimized arrangement is compared with Fig. 6.40, one observes, that the use of the optimum arrangement results in a more symmetric and homogeneous growth of the frost body as compared to an equidistant arrangement of the pipes.

Fig. 6.42

Comparison of the optimum arrangement of freeze pipes with an equal distribution of the pipes for $\mathbf{v}_{\text{L}}=1.0$ m/d, [51], [50]

Fig. 6.43

Influence of seepage flow on the formation of a frozen arch for an optimized arrangement of the freeze pipes, [51], [50]

A development of a failure criteria based on strength upscaling for freezing soils

A novel constitutive model for freezing soils is developed by adopting the CASM [93] for the unfrozen state, and the enhanced BBM [58] together with the homogenized strength criteria obtained for the freezing state [96], [97]. The developed elasto-plastic mechanical constitutive model with failure criterion upscaled through strength homogenization is named as Extended BBM [94]. In contrast to phenomenological elasto-plastic models, the failure criteria is established on the basis of the knowledge of their microstructure, i.e. the volume fractions and strength properties of constituent phases. Therefore, the macroscopic strength properties of drained partially frozen soil obtained through a two-step strength upscaling are incorporated into the extended BBM. Figure 6.44 illustrates the flowchart of the extended BBM with the strength properties from the strength upscaling model and to its integration into the three-phase finite element model for soil freezing.

Fig. 6.44

Illustration of AGF simulation under seepage flow considering the developed extended BBM with strength properties from a two set up-scaling strategy, [94]

6.4.2 Model Generation and Simulation Procedure

An automatic modeler which can be integrated within the TIM has been developed to generate realistic three-dimensional tunnel simulation models much simpler than up to now. Details of this approach along with an outlook to possible parallelization are given in the following sections.

6.4.2.1 Automatic BIM-Based Model Generation

Generating a realistic three-dimensional tunnel simulation model that is capable of representing all the involved components of a bored tunnel construction process requires considerable effort and experience [7]. In addition to the considerable experience required to generate an accurate numerical model, data from many different sources is required for the construction of such a simulation model. This data is most often not centrally stored and therefore not easily accessible. Additionally, the formats in which project data are stored are typically not compatible with the formats required by an FE program. This is especially true in the case of geometrical data, such as CAD drawings. Although existing design drawings are most often used as the basis for an FE model, a direct import into an FE program is most often not successful as the imported data generates a geometry that does not fulfill the requirements of the FEM model, such as model connectivity. Even if the geometry is successfully imported, other necessary aspects of an FE model, such as boundary conditions and material properties, must be applied manually. These incompatibilities inevitably result in the creation of a new model, which, especially in the case of complex 3D models, is a time consuming process.

To this end, an automatic modeler, which is integrated within the TIM, has been developed as a numerical simulation tool to automate and simplify the modeling process, and to reduce errors due to inexperience. This modeler allows for ‘‘BIM-to-FEM’’ interaction platform, that automatically extracts the relevant information (geology, alignment, lining, material and process parameters), needed for an FE-simulation from TIM sub-models and subsequently performs an FE-analysis of the tunnel drive. All process data and material parameters are stored and can be exported as text files. Moreover, for all geometrical representation, each individual component is defined as volumetric data with a non-uniform rational basis spline (NURBS) in ACIS^® .sat format. The necessary boundary conditions and construction sequences are automatically incorporated based upon the design data. This simplifies the information flow and limits errors that jeopardize the structural analysis.

The modularity of the developed TIM-based modeler has been accomplished by connecting various scripts and software packages, to which a specific task during the model generation is determined. The main involved modules are the GiD pre- and post-processing software, the object-oriented finite element framework KRATOS and library of auxiliary scripts (e.g. Python) used to invoke the TIM, exchange data and generate the complete simulation model. The modular structure of GiD allows automatic model generation using batch files and user-defined routines, written in TCL. These TCL-routines are powerful tools as they can access the data structure of GiD and automatically identify layers and boundaries without any user interactions. On the other hand, the batch files contain every instruction that is necessary to read and create the geometry, assign the boundary conditions and generate the mesh. Furthermore, it has the capability of invoking GiD in silent mode from command line. For this reason, the automatic creation of a number of batch files, based on the input parameters from the TIM is the key feature which simplifies and automates the complete process of model generation. Figure 6.45 outlines the workflow of the proposed automatic modeler. The main tasks that are performed during the automatic model generation are described below:

Fig. 6.45

Schematic illustration of the workflow of TIM-based automatic model generation with performed individual task

• Acquisition of geometry and geology: The topology of the geological layers, buildings and all relevant components of the tunnel geometry, in addition to the geometrical boundaries (simulation domain) must be defined. This information is imported from the TIM to the modeler as volumetric data with a non-uniform rational basis spline (NURBS) in ACIS^® .sat format. ACIS is a geometric modeling kernel used to enable robust and 3D modeling capabilities of Computer-aided design (CAD) software. The tunnel geometry can be either imported using the geometrical data for each individual ring, or as simply a tunnel alignment and a ring thickness and diameter. Initially the modeler stores these geometries separately.

• Acquisition of input parameters: Process parameters (e.g. face support pressure and grouting pressure) and material parameters (i.e. for soil, grouting, and lining stiffness) are required to generate the simulation model. This information is available in the TIM and can be provided in .txt format.

• Generation and execution of the numerical simulation: A set of batch files will be automatically generated after reading all required data (.txt). These batch files, import all geometrical information from 3D interchange files (ACIS) provided by TIM, generate the geometrical model in GiD, apply boundary conditions and material properties to their corresponding elements and generate the finite element mesh, automatically. Finally, a python simulation script is automatically generated, reading all process parameters from the TIM, and executed in order to perform the finite element analysis.

• Transmission of the simulation results to the Tunnel Information Model: The modeler is able to automatically incorporate the results of FE simulations into the TIM where the user has continuous access. The simulation data is thus easily visualized and compared to site data, such as building locations, monitoring data, such as settlement measurements, and machine data such as thrust forces.

The object oriented architecture of the computational software ekate and the modularity of the developed automatic modeler, allow for an efficient and robust internet-based interaction with the TIM via web-service.

6.4.2.2 Parallelization Concept

To speed up the computation process, the tunnel advancement analysis can be solved in parallel on a distributed computing platform using the Message-Passing-Interface (MPI). In the first stage, the mesh is decomposed into partitions to support for parallel assembly. As a result, the linear system has to be stored in parallel using a compatible storage scheme [4]. In the second stage, the resulting linear system is solved using a parallel iterative solver. To speed up the convergence of the iteration process, a block preconditioning strategy, taking advantage of the coupled nature of the fully saturated soils problem at hand, is employed.

The domain decomposition algorithm minimizes the interface between the domain to save communication time [40]. An illustration of a typical partitioning for tunnel analysis can be found in Fig. 6.46, left. To accommodate for contact analysis, where the TBM traverses through the computational domain and hence is in contact with the interface, the master contact surfaces are duplicated in all domains as ghost entities. It helps to compute the contact forces properly and avoid sophisticated data exchange, meanwhile the memory overhead is not high. An illustration for this concept is depicted in Fig. 6.46, right.

Fig. 6.46

The domain decomposition concept for tunnel analysis: (left) partitioning of the computational model; (right) partition of one domain with contact faces

The parallel iterative solver is GMRES, which is a variant of the Krylov subspace method. In this method, the solution is searched in the subspace $\mathcal{S}=\{\mathbf{x}_{0},\mathbf{K}\mathbf{x}_{0},\ldots,\mathbf{K}^{m-1}\mathbf{x}_{0}\}$, in which $m$ is the dimension of the Krylov subspace, $\mathbf{K}$ denotes the system stiffness matrix and $\mathbf{x}_{0}$ is the initial prediction. To speed up the search iteration, a preconditioner is required. For a fully saturated model comprising solid and fluid phases, the stiffness matrix $\mathbf{K}$ can be written in the block form $\mathbf{K}=\begin{bmatrix}\mathbf{A}&\mathbf{B}_{1}\\ \mathbf{B}_{2}&\mathbf{C}\end{bmatrix}$. This allows to compute an approximation for the inverse of $\mathbf{K}$ based on algebraic decomposition. This approximation can then be used as a preconditioner for the Krylov subspace method. The full algebraic factorization of the block system matrix reads

$$\displaystyle\mathbf{K}^{-1}\approx\mathbf{P}_{\text{sf}}^{-1}=\begin{bmatrix}\mathbf{I}&-\mathbf{A}^{-1}\mathbf{B}_{1}\\ \mathbf{0}&\mathbf{I}\end{bmatrix}\begin{bmatrix}\mathbf{A}^{-1}&\mathbf{0}\\ \mathbf{0}&\mathbf{S}^{-1}\end{bmatrix}\begin{bmatrix}\mathbf{I}&\mathbf{0}\\ -\mathbf{B}_{2}\mathbf{A}^{-1}&\mathbf{I}\end{bmatrix}.$$

(6.39)

In Eq. 6.39, $\mathbf{S}$ is the Schur complement of the system matrix $\mathbf{K}$, which can be approximated by $\mathbf{S}=\mathbf{C}-\mathbf{B}_{2}\,\text{diag}(\mathbf{A})^{-1}\mathbf{B}_{1}$. The block preconditioner Eq. 6.39 contains two matrix inversion $\mathbf{A}^{-1}$ and $\mathbf{S}^{-1}$. These inversions can be approximated by using classical preconditioning techniques, such as Incomplete LU factorization (ILU) or the Algebraic Multigrid Method (AMG). The preconditioner Eq. 6.39 can be simplified by dropping the first or last term, leading to the block lower and block upper preconditioner [13], as described in the equations

$$\begin{aligned}\mathbf{P}_{\text{sl}}^{-1} & =\begin{bmatrix}\mathbf{A}^{-1}&\mathbf{0}\\ \boldsymbol{O}&\mathbf{S}^{-1}\end{bmatrix}\begin{bmatrix}\mathbf{I}&\mathbf{0}\\ -\mathbf{B}_{2}\mathbf{A}^{-1}&\mathbf{I}\end{bmatrix}\Biggl(=\begin{bmatrix}\mathbf{A}&\mathbf{0}\\ \mathbf{B}_{2}&\mathbf{S}\end{bmatrix}^{-1}\Biggr),\end{aligned}$$

(6.40)

$$\begin{aligned}\mathbf{P}_{\text{su}}^{-1} & =\begin{bmatrix}\mathbf{I}&-\mathbf{A}^{-1}\mathbf{B}_{1}\\ \mathbf{0}&\mathbf{I}\end{bmatrix}\begin{bmatrix}\mathbf{A}^{-1}&\mathbf{0}\\ \mathbf{0}&\mathbf{S}^{-1}\end{bmatrix}\left(=\begin{bmatrix}\mathbf{A}&\mathbf{B}_{1}\\ \mathbf{0}&\mathbf{S}\end{bmatrix}^{-1}\right).\end{aligned}$$

(6.41)

Alternatively, to avoid computing the Schur complement, one can employ the algebraic Gauss Seidel decomposition of the block stiffness matrix $\mathbf{K}$,

$$\displaystyle\mathbf{P}_{\text{gs}}^{-1}=\begin{bmatrix}\mathbf{I}&\mathbf{0}\\ \mathbf{0}&\mathbf{C}^{-1}\end{bmatrix}\begin{bmatrix}\mathbf{I}&\mathbf{0}\\ -\mathbf{B}_{2}&\mathbf{I}\end{bmatrix}\begin{bmatrix}\mathbf{A}^{-1}&\mathbf{0}\\ \mathbf{0}&\mathbf{I}\end{bmatrix}.$$

(6.42)

The parallel computation phase in the context of the whole computational workflow for tunnel simulation is depicted in Fig. 6.47.

Fig. 6.47

Computational workflow for parallelized tunneling simulations. The parallel computation is highlighted in red, including possible block preconditioning strategies [13]

The domain decomposition strategy is crucial to obtain the perfect parallel scalability of the assembly process. Meanwhile, the performance of the block preconditioner varies and shows a degree of sensitiveness with regards to the permeability of the fluid phase. It is noted that the system stiffness matrix becomes ill-posed when the permeability is low, leading to a poor convergence of the iterative solver. The numerical experiment shows that the computational time is higher with low permeability, nevertheless the speed-up is better, as can be seen in Fig. 6.48.

Fig. 6.48

Performance of the block preconditioner with regards to high water permeability (left) and low water permeability (right)

6.4.3 A Tunnel Analysis Model Based on the CutFEM Method

To accommodate for the flexibility in varying the tunnel alignment in analysis and design, a tunnel analysis model based on the Cut Finite Element Method (CutFEM) is proposed. In this section, details of the CutFEM method along with the validation examples are presented. By the end of the section, a numerical example to analyze the induced building damage with regards to different tunnel track variants is performed to verify the effectiveness of the CutFEM approach.

6.4.3.1 Description of the Computational Model

The CutFEM tunnel model makes use of the fictitious domain concept [24], where the computational domain covers up the whole underground space, including the excavation domain. The material removal during excavation is accounted for in the weak form of the boundary value problem, which is then translated into the system stiffness matrix and residual forces. An illustration of the computational domain can be found in Fig. 6.49.

Fig. 6.49

The fictitious domain concept for tunnel analysis: The physical domain $\Omega_{\text{phys}}$ is extended by the fictitious domain $\Omega_{\text{fict}}$ to allow structured meshing of the computational domain $\Omega$. The indicator function $\alpha$ implicitly defines $\Omega_{\text{phys}}$ and $\Omega_{\text{fict}}$

In making up the computational domain $\Omega$, the fictitious domain $\Omega_{\text{fict}}$ is perceived as an extension of the physical domain $\Omega_{\text{phys}}$. As a result, the computational domain does not contain any geometric constraint and can be meshed arbitrarily. However, it is typical in practice to have structured meshing for simplicity and avoid remeshing. The structured mesh also offers other advantages, such as superior mesh quality and optimal assembly kernel for large simulation.

The indicator function $\alpha$ is defined as $\alpha\left(\mathbf{x}\right)=1,\forall\mathbf{x}\in\Omega_{\text{phys}}$ and $\alpha\left(\mathbf{x}\right)=0,\forall\mathbf{x}\in\Omega_{\text{fict}}$. The indicator function is also used to penalize the stress in the fictitious domain. For a typical two-phase fully saturated soil, under quasistatic conditions and assuming small strain kinematics, the equilibrium equations read

$$\begin{aligned}\delta W^{s} & =\int_{\Gamma^{s}_{N}}\delta\mathbf{u}^{s}\cdot\mathbf{t}dA-\int_{\Omega}\alpha^{s}(\mathbf{x})\nabla^{sym}\delta\mathbf{u}^{s}:\boldsymbol{\sigma}\mathrm{d}V+\int_{\Omega}\alpha^{s}(\mathbf{x})\delta\mathbf{u}^{s}\cdot\rho\mathbf{g}\mathrm{d}V=0,\end{aligned}$$

(6.43)

$$\begin{aligned}\delta W^{w} & =\int_{\Gamma^{w}_{N}}\delta p^{w}q^{w}dA-\int_{\Omega}\alpha^{w}(\mathbf{x})\mathop{\mathrm{grad}}\delta p^{w}\cdot\tilde{\mathbf{v}}^{ws}\mathrm{d}V+\int_{\Omega}\alpha^{w}(\mathbf{x})\delta p^{w}\,\text{div}\,\dot{\mathbf{u}}^{s}\mathrm{d}V=0,\end{aligned}$$

(6.44)

in which $\alpha^{s}$ and $\alpha^{w}$ are separate indicator functions for solid and fluid phases. In practice, the indicator function takes small values ($10^{-4}\sim 10^{-15}$) at the fictitious domain to avoid ill-conditioning.

The weak forms Eq. 6.43 and Eq. 6.44 are typically integrated using an adaptive quadrature scheme, see e.g. Fig. 6.50, where the integration points are generated by a space-tree subdivision algorithm. The adaptive quadrature scheme is simple to implement and robust with regards to a large number of material models.

Fig. 6.50

The adaptive quadrature for CutFEM tunnel analysis

To improve the mesh resolution around the excavation domain, and especially to avoid costly global remeshing, the structured mesh can be refined by using a space-tree subdivision approach. This approach involves the hanging node, which can be handled by introducing additional constraints to the resulting linear system, as illustrated in Fig. 6.51.

Fig. 6.51

Hanging nodes in adaptive mesh refinement (red dot: displacement node; blue dot: pressure node; red square: hanging displacement node; blue square: hanging pressure node) (left) and the relative location of the refinement domain containing hanging nodes with respect to the tunnel lining (right)

The fictitious domain approach also allows to mesh the components independently. In the context of tunnel analysis, the tunnel lining and the grouting layer can be meshed in a structured way as a tube mesh, see Fig. 6.52. The tube mesh is then connected to the background mesh using a mesh tying technique, which effectively enhances the weak form by weak tying constraints

$$\begin{aligned}\delta W^{ut} & =\int_{\Gamma_{s}}\kappa_{u}\left(\delta\mathbf{u}_{1}-\delta\mathbf{u}_{2}\right)\cdot\left(\mathbf{u}_{1}-\mathbf{u}_{2}\right)\,\mathrm{d}A,\end{aligned}$$

(6.45)

$$\begin{aligned}\delta W^{wt} & =\int_{\Gamma_{s}}\kappa_{p}\left(\delta p_{1}-\delta p_{2}\right)\left(p_{1}-p_{2}\right)\,\mathrm{d}A.\end{aligned}$$

(6.46)

Fig. 6.52

Boundary fitted discretization of the tunnel lining (red) and the grouting layer (blue) (tube mesh). $\Gamma_{s}$ denotes the soil surface region, aka integration domain for tying constraints

The simulation of sequential excavation is characterized by continuously adapting the cut elements with the new cut configuration. The cut configuration is changed in each excavation step. There are two possibilities to realize the cut configuration, either using a Boundary Representation (BRep) of an excavation tube or a level set based on the tube BRep. The BRep approach is preferred in practice due to its simplicity, nevertheless the level set approach could provide more options to fine-grain adjust the soil surface. A typical cut changing situation is depicted in Fig. 6.53 and the resulting quadrature in 3D is depicted in Fig. 6.54.

Fig. 6.53

Cut element (red) changes from partially cut to fully cut during tunnel advance

Fig. 6.54

Soil surface (light blue), heading face (purple) and quadrature points in the physical domain (green) evolving during sequential soil excavation

6.4.3.2 Numerical Examples and Validation

The results of the CutFEM tunnel model is shown to match excellently with the analysis using FEM [14], as can be seen in Fig. 6.55, where the straight advancement of a TBM with an overburden of $2D$ ($D=9.5$ m) is simulated. The analysis with CutFEM, however, takes more computational resources since the refinement is performed excessively around the tunnel track, leading to more elements. In addition, between excavation steps, CutFEM simulation requires to transfer the variables to the newly cut cells to maintain the stress accuracy.

Fig. 6.55

Simulation of mechanized tunnel advance with CutFEM: (Left) Comparison of the settlement profile at line $L$ at two advancement stages obtained from the finite element and the CutFEM model; (Right) Bending moment on middle ring of the tunnel

Having been fully validated, the CutFEM model shows a great potential for parametric study and tunnel track design. Figure 6.56 shows an example to evaluate the building damage during excavation for different tunnel track variants. For all analyses, the same background mesh is used for mesh and quadrature refinement. Thanks to the mesh refinement process based on space-tree subdivision, which is highly efficient, the only change in the input given to the analysis is the tunnel track data, characterizing by sampling points along the alignment curve.

Fig. 6.56

CutFEM models used to investigate design variants in urban tunneling: Top: Background mesh for three track variants, bottom: Visualization of building damage assessment

6.5 Risk Assessment of Building Damage

Besides the classical damage assessment needed during TBM operations, research was done to determine specific coefficients required for the possible consideration of compensation injections. To aid the assessment, an interactive tool was developed and its usefulness is demonstrated by applying it to a case study involving the Wehrhahn line in Düsseldorf.

6.5.1 Classical Damage Assessment

Damage assessment of buildings is usually based on the category of damage (cod), see e.g. [27, 56, 92]. In this work, the tunneling induced building damage is quantified by comparing the maximum of the calculated structural strains with limiting strains, which – in the case of brittle materials such as concrete or masonry – lead to either no damage, micro-cracking or macro-cracking [48, 59, 62]. Table 6.1 shows a common assignment of limiting tensile strains to corresponding categories of damage. While strains in category 0 cause no damages, categories 1–2 cause usually aesthetic or optical damages, strains in category 3 impair structures’ serviceability and strains in category 4–5 even affect the structures’ ultimate load bearing capacity [12, 61].

Tab. 6.1 Relationship between category of damage and limiting tensile strains

6.5.1.1 Settlement Prediction

The damage-causing event, the settlements can be determined using analytical models like Peck [67] and Attewell [3]. Here the focus was set on the aforementioned settlement prediction using 3D FEM calculations.

6.5.1.2 Damage Evaluation

Based on settlement values for the transversal and longitudinal direction the bending strains $\varepsilon_{b}$ and shear strains $\varepsilon_{d}$ for the hogging (hog) and sagging (sag) areas are obtained by using Eqs. 6.47–6.50, [16]. The individual settlements are referred to as $\Updelta_{i}$, where the index $i$ denotes the settlement area hog or sag. In addition to the settlements, the corresponding lengths $L$ for the hogging $L_{\text{hog}}$ and sagging area $L_{\text{sag}}$ are taken into account. The height $H$ of the building corresponds to the distance from the foundation to the eaves. The $E/G$-ratio represents the continuum mechanical relationship for a linear-elastic material. $E$ terms the Young’s and $G$ the shear modulus. Influences such as facade openings and degradation effects, which reduce the effective stiffness, can be considered in fictitious E/G values. For ordinary masonry structures with a low tensile strength, $E/G$ equals 0.5 [16] and we have

$$\begin{aligned}\varepsilon_{b,\text{max},\text{hog}} & =\dfrac{\Updelta_{\text{max},\text{hog}}}{L_{\text{hog}}}\cdot\left(\dfrac{L_{\text{hog}}}{12H}+\dfrac{H}{2L_{\text{hog}}}\cdot\dfrac{E}{G}\right)^{-1},\end{aligned}$$

(6.47)

$$\begin{aligned}\varepsilon_{d,\text{max},\text{hog}} & =\dfrac{\Updelta_{\text{max},\text{hog}}}{L_{\text{hog}}}\cdot\left(1+\dfrac{L_{\text{hog}}^{2}}{6H^{2}}\cdot\dfrac{G}{E}\right)^{-1},\end{aligned}$$

(6.48)

$$\begin{aligned}\varepsilon_{b,\text{max},\text{sag}} & =\dfrac{\Updelta_{\text{max},\text{sag}}}{L_{\text{sag}}}\cdot\left(\dfrac{L_{\text{sag}}}{6H}+\dfrac{H}{4L_{\text{sag}}}\cdot\dfrac{E}{G}\right)^{-1},\end{aligned}$$

(6.49)

$$\begin{aligned}\varepsilon_{d,\text{max},\text{sag}} & =\dfrac{\Updelta_{\text{max},\text{sag}}}{L_{\text{sag}}}\cdot\left(1+\dfrac{2L_{\text{sag}}^{2}}{3H^{2}}\cdot\dfrac{G}{E}\right)^{-1}.\end{aligned}$$

(6.50)

All four equations must be evaluated if the structure is located in both, in the hogging and sagging area, only the corresponding ones for structures located in either of the two. Furthermore, the strains and the associated damage category have to be determined for each individual structure in transversal and longitudinal direction. To account for a variation from the orthogonal orientation of the buildings, the settlement values $\Updelta_{i}$ are determined for the projected facades separately for the longitudinal and transversal direction (see Fig. 6.57). The related length $L_{i}$ ($i=\text{sag},\,\text{hog}$) is taken from the building rotated by the angle $\theta$. This means that for wall $AD$ with length $L_{\text{sag}}$ the damage caused by the settlement $\Updelta_{DA,\text{sag}}$ in transversal and $\Updelta_{AD,\text{sag}}$ in longitudinal direction must be determined (Fig. 6.57). The maximum value of the strains is then relevant for the evaluation.

Fig. 6.57

Diagonal excavation; side view in transversal (bottom left) and longitudinal direction (top right); top view (bottom right)

In contrast to the damage assessment in the final state, which assigns single damage to each location of the alignment, the potential damage during the construction process depends on the distance of the building from the tunnel face [28]. The distance that provides the maximum strain and corresponding maximum damage is determined by incrementally changing the location of the building concerning the settlement trough.

However, it is inefficient to evaluate the damages by using equidistant increments along a whole alignment. On the one hand, this requires a large number of calculations, and, on the other hand, only the maximum damage is usually relevant for each building. An approach based on Artificial Neural Networks (ANN) is derived to determine these maximum damages. The advantage of using an ANN is that any type of building or soil model (analytical or numerical) can be used [18].

ANNs try to approximate data of underlying problems by a combination of simple sub-functions, called neurons. The impact of an input parameter on a certain neuron is determined by optimization that delivers weights as scalar quantities. An arbitrary number of neurons is vertically summarized on a single layer. More layers are optional. The greater the number of neurons or layers, the better the network fits to the data, but the lower the forecast quality of new data is [36]. ANN has often been proved a useful alternative to multiple polynomial regression since it neither is too demanding concerning the quality of input data nor the residuals of approximation. Basically, a variety of ANN’s exist which differ regarding the topology of the network and interconnections therein. Most frequently used for approximation, are simple feed-forward networks [1]. To establish an ANN, the total available data is split into three portions: training, validation, and test data.

• Training data serves to determine optimal weights by iteration.

• Validation data provides an indicator for optimal weights. As long as the error of validation data decreases, the weights are improved based on the training data. If the error increases over 6 iterations, training is canceled and the weights associated to the minimal error are used [31].

• Test data finally helps to prove the forecast ability of the ANN [59].

The used ANN consists of one hidden layer with 10 neurons [18, 60] and has seven input variables. These are the building length $L$, building height $H$, the rotation angle $\theta$, the distance of the building from the tunnel axis in transversal direction $e_{y}$, the dimensionless soil parameter $K$, the coverage $c$, and the external tunnel diameter $D$. The input parameters were chosen for typical urban environments (see Fig. 6.58, above) and generated as uniformly distributed data using the Latin Hypercube Sampling. The eccentricity at which the maximum strain is obtained was used here as the output. Based on the eccentricity value, the strain can then exactly be determined in a subsequent calculation. For computationally expensive models (e.g. FE models), however, it is more appropriate to use the strain itself as the output parameter since this eliminates the need for a subsequent FE calculation.

Fig. 6.58

Comparison of the exactly determined eccentricity e and the prediction based on the ANN for training (left), validation (center) and test data (right)

To generate the ANN’s, 120,000 input, 15,000 validation, and 15,000 test data were used. Figure 6.58 shows the comparison between the exact and predicted eccentricity $e$ of the ANN for training (left), validation (middle), and test data (right). In the optimal case, the results of both calculations would lie on a diagonal regression line. Data points above the regression line indicate an overestimation, and those below an underestimation of the eccentricities by the ANN. A scalar quantity for the characterization of the prognosis quality is the coefficient of determination $R^{2}$. The coefficient is determined concerning the diagonal regression line and is about 0.98, which means that the model has a very high prognosis quality. To evaluate the error and the prognosis time additional eight test data (Fig. 6.58 right, green circles) were used. The prognoses were calculated on average in about 0.1 second and showed a mean deviation of 0.43%. In conclusion, it can be stated that based on the ANN, the location of the maximum strains and the duration of the prognoses are sufficiently accurate for real-time predictions [19, 54].

6.5.2 Advanced Damage Assessment Coefficients

Classical damage assessment is based on mechanical damage only, but does not adequately reflect the background of actual use, the related consequences on maintenance and restrictions of use, and the effects of damage-reducing compensation measures.

Therefore, we propose an approach that combines the mechanical damage, i.e. the categories of damage, with the maintenance costs as well as building-specific properties such as the type of the building and a possible consequence of the damage (Fig. 6.59, left). To compare the maintenance and the prevention of damages, evaluation coefficients for possible consideration of compensation injections are also introduced (Fig. 6.59, right). This concept enables a detailed comparison of maintenance and prevention of damage and can be used to determine the most economical approach during the design of urban tunnel alignments (Fig. 6.59, middle).

Fig. 6.59

Maintenance coefficients (left), risk assessment (middle), compensation measures (right)

6.5.2.1 Effort Coefficients for the Maintenance of Buildings

The transfer of theoretical limiting-tensile-strain-based cods to more practical effort coefficients builds on the damage descriptions of Table 6.1 and statistical maintenance costs for residential buildings according to the German Baukosteninformationszentrum [5] (BKI). Naturally, this data basis depends on regional costs and construction methods, as well as highly subjective criteria regarding the demand for maintenance. These criteria should therefore be adapted to regional preferences. The determination of the damage assessment coefficients is a simple additive concept. With increasing cod, i.e., increasing damage, maintenance becomes more elaborate. In the case of cod 1, for example, only painting work is required to restore the building’s original condition. For cod 2, the cod 1 painting works are complemented by more extensive maintenance like the backfilling of cracks. This addition of required measures continues up to cod 4. For cod 5 the structure is damaged beyond repair, and a new, equivalent construction is needed. Based on the considered reference project Wehrhahn-Line (WHL), maintenance measures are assumed as follows:

• cod 0: No measures required

• cod 1: Wallpapering, painting and varnishing work

• cod 2: Interior wall maintenance

• cod 3: Plastering, maintenance, and sealing work

• cod 4: Maintenance of supply and disposal pipelines, exterior walls, and exterior wall lining

• cod 5: Build a new building

Due to the minor damage in cod 1, only painting work is assumed here as maintenance measures. With reference to Table 6.1, the main damage to be expected for cod 2 is at the interior walls, and therefore only this area needs to be maintained. As cracks of up to 5 mm are to be expected in exterior walls of cod 3, plastering work as well as necessary auxiliary constructions, like scaffolds, are required to carry out the maintenance of the structure. Since water tightness is no longer guaranteed, appropriate sealing work is required. The measures to remove jammed windows and doors are covered by the maintenance work. In particular, higher levels of damage can lead to the need to replace masonry and the corresponding wall lining. Large settlement differences can also damage supply and disposal lines, which must be replaced. Based on these maintenance measures, the relative evaluation coefficients, also known as effort coefficients, can be derived by means of the statistical BKI costs, which are usually provided per m${}^{2}$ of a building’s gross floor area (GFA). Since the estimated costs also largely depend on the standard of living, i.e., whether, for example, simple wallpaper or a complex filling of a wall is required, they are usually provided in a range of minimum, mean, and maximum values (Table 6.2). In Table 6.2, the mean value of a new building serves as the reference value (1.0) for the normalized effort coefficients $\alpha_{i}$. Consequently, maintenance for a building with cod 1 is $1/0.07=14$ times more economical on the mean value level than replacing it with a new structure.

Tab. 6.2 Costs and effort coefficients $\alpha_{i}$ depending on the category of damage

6.5.2.2 Building Coefficients Based on Raw Construction Values

Although the effort coefficients provide a generalized maintenance demand with respect to the cods, the specific type of building is still disregarded. To also incorporate this information, additional building coefficients $\gamma$ are introduced, which take the varying construction expenses of different building types into account. They represent generalized, relative raw construction costs per m${}^{3}$ of the building volume. For the WHL reference project they are defined according to the values provided by the Ministerial Bulletin of the State of North Rhine-Westphalia, Germany (NRW 2018/27). Table 6.3 presents some relevant average raw construction costs and their corresponding normalized building coefficients. The value of a residential building serves as the reference. If buildings have higher raw construction costs, the building coefficient increases, whereas it decreases for lower costs.

Tab. 6.3 Building coefficients depending on selected raw construction costs

6.5.2.3 Consequence Coefficients

Although a damage assessment which incorporates building and effort coefficients is already much more accurate, the additional consideration of a building’s restricted use after damage is required for a holistic determination of damage consequence costs. Depending on the type of use and duration of the maintenance measures, these impairments can have a considerable influence on the total damage costs. For these consequential costs, however, no publicly accessible, taxable data basis is available. For this reason, here, the influence is estimated by means of deterministic consequence coefficients. They are indicated as a function of the type of building or use, which is classified analogously to the consequence classes (CC) of the VDI Guideline 6200 (2010) (Table 6.4).

Tab. 6.4 Consequence coefficient in relation to the consequence class

CC 2 is used as reference value. Its corresponding consequence index is set to 1.0, i.e. the restricted use costs are equal to the maintenance costs. For CC 1 they only add up to 50 %, for CC 3 to 150 % of the maintenance costs. The final assessment coefficient $S$ of the damage of a building regards maintenance measures as well as the damage consequences and corresponds to the product of the effort coefficient $\alpha_{i}$, building coefficient $\gamma$, consequence coefficient $\phi_{k}$ as well as the total building volume $V$ (Eq. 6.51)

$$\begin{aligned}S={\underbrace{\alpha_{i}\cdot\gamma\cdot V}_{\text{maintenance}}}+{\underbrace{\phi_{k}\cdot\alpha_{i}\cdot\gamma\cdot V}_{\text{consequence of damage}}}=(1+\phi_{k})\cdot\alpha_{i}\cdot\gamma\cdot V\quad\text{for}\quad\begin{cases}i=0,\dots,5\\ k=1,\dots,3\end{cases}\end{aligned}$$

(6.51)

6.5.2.4 Reduction of Settlement Induced Damages

The effort of repairing damages according to Eq. 6.51 is opposed by that of avoiding damage by preemptive measures. Depending on the strategy and procedure, different methods for the avoidance can be applied. On the one hand, these are reinforcing measures of the structures, e.g. supporting the facades or reinforcing the foundations, and, on the other hand, a reduction of the expected settlements. These reductions can be achieved, for example, by increasing the tunnel’s coverage, actively controlling the excavation (jack forces, annular gap grouting) or using compensation measures.

Here, only compensation measures are regarded, which can be temporary or permanent. Temporary measures include, for example, icing of the soil, whereas permanent measures include, for example, high-pressure mortar injections (HPI). HPIs inject mortar into the ground under high pressure to locally strengthen the ground. The total effort of realizing HPIs, i.e. including mounting and dismounting of the equipment, man power, material, etc., is usually stated per m${}^{3}$ of mortar injected. It strongly depends on the condition of the soil as well as various environmental conditions. Experience values for standard urban measures lie between 400 and 600 €/m${}^{3}$. This corresponds to a compensation factor $\kappa$ of 400/129 = 3.10 to 600/129 = 4.65 based on the reference value, the building volume of a residential building. The total cost of an injection $K$ is thus determined by the product of the compensation factor $\kappa$, the thickness $h$ of the injection and the base area of a building $A$,

$$\begin{aligned}K=\kappa hA\leavevmode\nobreak\ .\end{aligned}$$

(6.52)

6.5.3 An Interactive Tool for Visualizing Building Damage

The visualization as well as the interactive design is implemented via the program Rhinoceros 3D (Rhino 3D) [71]. The required algorithms were created using the associated, visual programming language Grasshopper 3D [70].

To implement the visualization, the building’s coordinates must first be available in a suitable file format. The building geometries and the position (distance and rotational angle) in relation to an arbitrary reference coordinate system can then be determined from the provided coordinates (Fig. 6.60, top left).

Figure 6.60 (bottom) shows the developed calculation routine for the visualization of the damages in Grasshopper 3D. First, the building data (blue box) and the tunnel information–outer diameter, depth and alignment (green box)–are imported. In a second step, the distances between the buildings and the tunnel axis are determined, followed by the damage evaluation in the final and construction state (red box). In a final step, the potential damages are transferred to Rhino 3D using the corresponding color code of the categories of damage (turquoise box).

Generally, for any given location of the tunnel, all buildings in the considered urban area have to be analyzed. In case of interactive design as well as in case of a large number of buildings, this leads quickly to a significant computational effort despite using analytical models. For this reason, only those buildings are included in the evaluation which are located at a distance smaller than $r=2.5K\left(c+D/2\right)$ from the tunnel axis. Therein $K$ is the dimensionless soil parameter, $c$ is the coverage, and $D$ is the external tunnel diameter (Fig. 6.60, top right). Investigations in [3] have shown that beyond this distance usually no relevant settlements occur.

During interactive design, the user can adjust the tunnel diameter, the depth and the alignment as desired. The alignment is modeled using non-uniform rational B-splines (NURBS). Corresponding constraint points or boundary conditions can be introduced during modeling using NURBS by means of corresponding support points. Furthermore, it is possible to shift an arbitrarily modeled alignment holistically.

Fig. 6.60

Building coordinates (top left); Computational boundaries (top right); algorithmic implementation in Grasshopper 3D (bottom)

6.5.4 Visualized Damage Assessment Illustrated by the Wehrhahn Line

The previously presented methods shall be applied to the reference project Wehrhahn-Linie in Düsseldorf, [49, 75]. The entire WHL has a length of $3.4$ km. Here, the east branch of the alignment, which is approximately $800$ m long, is used for visualization. Data on 285 buildings are available along this east branch. The tunnel was excavated at a depth of about $12.5$ m with an external diameter of 9.2 m in loose to medium dense sands and gravels.

6.5.4.1 Application of Damage Assessment Coefficients

112 structures are located in the vicinity of the WHL tunnel and are evaluated using the aforementioned damage assessment coefficients. Figure 6.61 (top) shows a summary of the maximum cod for all 112 structures. Clearly, a direct underpass of the facade leads to higher damages which decrease with increasing eccentricity. In total, there are 16 out of 112 vulnerable buildings. All of them have a cod $>$ 0.

Following a basic plausibility check of the prognosis, the risk analysis is performed. This analysis serves on the one hand to determine the occurrence or exceedance probability of potential efforts for damage maintenance, and on the other hand to determine the effort and benefit of potential preemptive settlement reducing measures under uncertain boundary conditions, such as compensation injections.

If the risk measures for the minimum, average and maximum maintenance efforts are determined successively for the 112 facades, the uncertain risk of damage can be represented as a function of buildings protected by compensation injections (Fig. 6.61, bottom left).

As expected, this damage risk has the highest maintenance effort at 0 protected structures. It decreases continuously with an increasing number of protected structures until it reaches a minimum effort of R = 0 for 16 protected structures. These 16 structures represent the critical buildings of Figure 6.61 (top) with cod $>$ 0. For each building removed from the risk simulation, the uncertain effort of the compensation injections increases. The positive effect of the compensations is recognizable up to 16 protected buildings, whereas for the remaining 96 buildings – which permanently are in cod 0 – the compensations only lead to an increased effort but negligible benefits. The sum of the two curves results in the uncertain total effort (Fig. 6.61, bottom right).

Fig. 6.61

Maximum expected category of damages for buildings along the WHL (top); Uncertain risk simulation: damage risk, compensation measures (bottom left) and total effort (bottom right) [59]

6.5.4.2 Deploying the Interactive Tool

Figure 6.62, (top left) shows the structural damage assessment of the WHL. In the analysis, possible diagonal, corner, and side excavations as well as the construction stages were taken into account. Basically, for all buildings located in the range $r=2.5K\left(c+D/2\right)$ to the tunnel axis, the damage in the final state as well as in the construction state is determined. However, only the maximum damage is shown in the visualization. In addition to the structural damage, the extended damage indices can also be visualized in order to obtain direct visual information on the maintenance effort for a building (Fig. 6.62, top right). Furthermore, a direct visualization of possible compensation measures as well as a comparison between compensation and maintenance measures is possible (Fig. 6.62, bottom).

Fig. 6.62

Predicted categories of damage for a possible alignment along the WHL (top left); maintenance as a function of the category of damage (top right); localization of suitable compensation injections (bottom left); visualization of the maintenance and the compensation injections (bottom right)

6.6 Interactive Assessment of Tunnel Alignment

The interactive platform uses information from the planning process to generate new data. Certain boundary conditions can be interpreted as geometry, such as a range constraint or reference points. However, this procedure can become much more significant if, for example, settlement analysis is taken into account.

6.6.1 Simulation Strategy for a Seamless Connection to TIM Models

For the assessment of the impact of various tunnel alignments, e.g., settlements, face stability, damage criteria, a process-oriented finite element (FE) simulation model for mechanized tunneling (ekate) is integrated in the interactive platform for tunnel track design. For this purpose, the FE-simulation model is coupled with the Tunneling Information Model (TIM) (see Fig. 6.63). This coupling allows automatic extraction of all relevant information (geology, alignment, lining, existing infrastructure, material, and process parameters) needed for the numerical model and subsequently performing a FE analysis for the tunnel drive.

Fig. 6.63

Automated creation and analysis of the FE simulation model ekate 2.0 based on TIM CAD-Models

6.6.2 Interoperability

In order to keep the TIM independent from the external applications, e.g. the simulation process, the TIM and the automatic modeler (introduced in Sect. 6.4.2) interact using a web-based service (see Fig. 6.64). This allows the TIM to autonomously invoke the simulation for a certain domain and automatically configure, generate and run the required three-dimensional finite element model. The web-based interface provides a simple interaction platform that allows any additional applications to access and exchange data in the TIM through predefined Model View Definitions (MVD’s) in a general manner. For the application of numerical simulation, the TIM can be invoked for a set of parameters that are required for the numerical simulation, which is read from the centralized database and automatically provided to the simulation.

Fig. 6.64

Schematic representation of TIM applications interacting through a web service which enables simulation software to automatically extract the required data and parameters in order to configure and generate the simulation model and perform simulations

6.6.3 Optimizing the Modeler for Various Tunnel Alignments

To further enhance the efficiency of model generation for various tunnel alignments, a new modeling strategy is integrated in the FE-simulation model that is flexible to incorporating various tunnel alignments without requiring generation of a new finite element mesh for each path. This new strategy employs CutFEM, which is based on a structured regular mesh of the entire domain and does not require the spatial discretization to follow the geometry of the (moving) excavation boundaries (see Fig. 6.65). The boundaries of the soil layers, the excavation geometries and the tunnel alignment are directly transferred as NURBS surface geometries in the ekate-software without any need for a new mesh generation. Adaptive Octree algorithms with quadrature refinement is used to realize geometric and material boundaries in the model. This procedure is a pre-requisite to enhance the interoperability with the interactive track design tool. On one the hand, the FE-simulation can be invoked autonomously for various tunnel alignments. On the other hand, the results of the numerical simulation are automatically incorporated in the digital planning tool.

Fig. 6.65

Automatic generation of Finite Cell simulation model for various tunnel alignments based on TIM CAD-Models

To enable real-time interaction with the platform, a surrogate modeling approach for various tunnel alignments can be trained beforehand based on synthetic data from FE-simulations and then integrated in the interactive platform.

6.6.4 A Surrogate Model For Real-Time Tunnel Alignment Assessment

The surrogate model is established and applied during the design stage of a tunnel project with a purpose of interactively investigating different possibilities of tunnel alignment by providing the associated surface settlement field in real-time. In the offline stage, i.e. before the interactive real-time design procedure, the process-oriented CutFEM model described in Sect. 6.4.3 is used to simulate the mechanized tunneling processes with various tunnel alignments. The resulting settlement fields will be used to construct the surrogate model. For each simulation, input parameters are coordinates of a set of points representing the tunnel alignment. In general, tunnel alignments can be in any shape satisfying the predefined design requirements. To have a parametric representation of a tunnel alignment, the alignment is assumed to be represented by a cubic function. As the first step to create an arbitrary alignment, some control points, which divide the tunnel alignment into several parts following the X direction, are generated using the Latin Hypercube sampling. A fitting step is performed to determine parameters of an appropriate cubic function which passes through these control points. Subsequently, coordinates of all input points are determined based on the fitted cubic function. These coordinates are then sent to the CutFEM model to execute the simulations as well as stored as the inputs of the surrogate model. Figure 6.66 presents inputs of the surrogate model for an illustrative example, which are possible tunnel alignments in three-dimensional space.

Fig. 6.66

a Possible three-dimensional tunnel alignments, b design constraints for the width of tunnel alignments, c design constraints for the depth of tunnel alignments

The objective of the surrogate model is to predict the complete surface settlement field for an arbitrary tunnel alignment with similar accuracy as the original CutFEM model. In the context of real-time applications, with respect to each intuitive alignment change, input points, which are well approximated with the new alignment, will be created using the curve fitting method. With the coordinates of the new input points, a surrogate model based on the combination of Proper Orthogonal Decomposition and Radial Basis Functions (POD-RBF) methods is employed to approximate the spatial field of surface displacements. The predicted results are used to support the design engineer in selecting appropriate alignments for further investigations. More details about the method can be found in [17, 66].

To illustrate the usage of the POD-RBF model, a representative example is performed, in which Fig. 6.67 depicts the very good agreement between the predicted settlement obtained from the described POD-RBF surrogate model with 2000 samples and the reference solution from CutFEM model for an illustrative validation alignment.

Fig. 6.67

Settlement prediction for a validation tunnel alignment scenario. a Tunnel alignment, b prediction (POD-RBF), c reference (CutFEM simulations)

Table 6.5 summarizes the $L_{2}$ norm prediction errors. One can see that, the prediction accuracy of the model is significantly increased when the data set is refined.

Tab. 6.5 Prediction performance of the POD-RBF surrogate model for three-dimensional tunnel alignment design example: $L_{2}$ norm error (in %)

The $L_{2}$ norm errors from 10 fold-cross validation prove the generalization and reliability of the surrogate model. While maintaining the prediction capability as compared to the CutFEM model, the surrogate model is able to considerably reduce the computation time from 6 to 10 hours to less than 1 second, Therefore, the POD-RBF surrogate model can substitute the CutFEM model for a real-time tunnel track design application.