Keywords

1 Introduction

In recent years, the simulation of technical systems has an increasing role from the concept phase up to the systems end of life. So called “digital twins” of real world systems allow the prediction of the machine behavior through simulations, allowing optimizations of the system during development and operation, model predictive control or predicting service intervals. Furthermore, time and money can be saved due to a significant acceleration of the development process. Different system concepts can be studied for the best suiting solution. Components can be investigated and optimized before building prototypes.

Unfortunately, using all available tools to develop modern systems, results in an steadily rising machine complexity. Systems from different areas like fluid components for temperature management, active bearings, sensors, actors and gears, etc., along with lots of control components that govern the interaction are incorporated into one machine.

To model the behavior of those complex systems, it is necessary to apply the principle of “divide and conquer” by modularizing the system model. Each individual module must represent its respective component with the application sufficient accuracy and finally, all those sub-modules have to be incorporated into the full system model.

The process of assembling the whole system model requires a standardized way to bring all the sub-models together and allow the simulation of the whole system. This is, unfortunately, a very difficult task, as there exist numerous proprietary and open source tools to facilitate different modeling techniques and approaches, which are best suitable in their respective field. A nowadays widely used tool to exchange models, even across software packages, is the standardized Functional Mockup Interface (FMI) [5]. It allows the easy combination of multiple system models through standardized interfaces for model inputs and outputs. As every tool, FMI has limitations, which will be addressed next.

To calculate the distribution of field quantities like the temperature or the displacement vector, partial differential equations (PDE) have to be solved. In most cases, no closed solution for those PDEs can be given for real world problems, since the systems geometries and governing equations are too complex. Therefore mathematical discretisation methods like the Finite Element Method (FEM), Finite Difference Method (FDM) or Finite Volume Method (FVM) have been developed to calculate approximate solutions to those field equations [6, 7]. Typically, the FEM is used to solve the here discussed thermoelastic problems.

Section 2 gives a short theoretical overview of the system equations resulting from the application of FEM to the governing PDEs. Model Order Reduction (MOR) as a tool to increase the simulation efficiency will be introduced. The system equations for thermoelastic problems are given.

Widely used proprietary CAE tools like Ansys® or Abaqus® cannot fulfill all modeling and simulation demands of the CRC/TR96, like model predictive control of machine tools by process parallel simulation or parameter optimization. An exchange format for PDEs with geometry information, similar to FMI, which also incorporates moving contact surfaces and loads, was necessary to facilitate the exchange and cooperation between the multiple specialized groups in the CRC/TR96. Efforts, mentioned in [9, 13] did not suffice the needs of the modeling goals.

Section 3 therefore will propose a toolchain developed in the CRC/TR96 to tackle the before mentioned model exchange and combination issues, using multiple developed data formats as interfaces between different software modules used to solve the modeling and simulation tasks.

After introducing all the tools, a comprehensive example for the usage of the toolchain is given by describing the modeling process for an in production machine tool. Different coupling approaches for the moving part contacts are discussed and exemplary simulation results are given.

The final section discusses the next steps necessary to facilitate the usage of the tools and proposed model in practical applications.

2 Theoretical Background

The derivation of equations to model physical phenomena on an infinitesimal small representative volume element, along with constitutive equations for the material behavior, typically results in a time and position dependent PDE. Those PDEs cannot be solved in a closed manner on the simulation domain in most cases, since the investigated problem is too complex. Approximation methods, become necessary to calculate an approximate field quantity distribution. One such method is the FEM, which will be introduced next.

FEM. The first step to apply the FEM consists of splitting the simulation domain into multiple small geometric segments, the finite elements. Each of those elements has edges and nodes on those edges. The entirety of nodes and elements of the domain is called the FE-mesh.

The real physical field, for example the temperature or the deformation, in each element is approximated by shape functions whose coefficients are the approximated field values at the nodes. Further mathematical transformations result in element matrices for every single finite element. Those element matrices can then be assembled together into the system matrices resulting in the following system of equations.

$$\begin{aligned} \boldsymbol{M} \ddot{\boldsymbol{\phi }}(t) + \boldsymbol{D} \dot{\boldsymbol{\phi }}(t) + \boldsymbol{K}\boldsymbol{\phi }(t) = \boldsymbol{r}(t) \end{aligned}$$
(1)

For demonstration purposes, it is assumed that all system matrices are time independent. The symbols \(\boldsymbol{M} \in \mathbb {R}^{n \times n}\), \(\boldsymbol{D}\in \mathbb {R}^{n \times n}\) and \(\boldsymbol{K} \in \mathbb {R}^{n \times n}\) represent the mass, damping and stiffness matrix of the system, \(\boldsymbol{\phi } \in \mathbb {R}^{n}\) is the system degree of freedom vector and \(\boldsymbol{r}\in \mathbb {R}^{n}\) denotes the right hand side vector.

Typically, those system matrices are sparse but have a very high dimension, which results in large simulation times, making those systems not applicable to the use as models for model predictive control. Furthermore, parameter studies and optimization runs can be very time consuming.

MOR can be a helpful tool in those circumstances, as it allows the reduction of the system degrees of freedom while preserving, the correct method provided, the chosen model outputs up to a certain accuracy.

MOR. The Eq. (1) generally can be rewritten into the state space form, further called input-output (IO)-Model.

$$\begin{aligned} \boldsymbol{E} \dot{\boldsymbol{x}}(t)&= \boldsymbol{A} \boldsymbol{x}(t) + \boldsymbol{B}\boldsymbol{u}(t) \end{aligned}$$
(2a)
$$\begin{aligned} \boldsymbol{y}(t)&= \boldsymbol{C} \boldsymbol{x}(t), \end{aligned}$$
(2b)

with the matrices \(\boldsymbol{E} \in \mathbb {R}^{n \times n}\), \(\boldsymbol{A}\in \mathbb {R}^{n \times n}\) and \(\boldsymbol{B} \in \mathbb {R}^{n \times p}\) \(\boldsymbol{C} \in \mathbb {R}^{m \times n}\), the system states \(\boldsymbol{x} \in \mathbb {R}^{n} \), the inputs \(\boldsymbol{u} \in \mathbb {R}^{p}\) and the model outputs \(\boldsymbol{y} \in \mathbb {R}^{m}\). The fact that every second order system can be rewritten as a first order system of double size is used.

By means of MOR it is possible to reduce the high-order IO-Model to a reduced order model (ROM). Typical methods to reduce the model order with projection based techniques are balanced truncation, moment matching and the iterative rational Krylov algorithm (IRKA) [3].

All those methods produce so called truncation matrices \(V, W \in \mathbb {R}^{n \times r}\) where the dimension r is much smaller than n. The matrices can then be used to generate a lower dimension state space model of the form

$$\begin{aligned} \boldsymbol{W}^T\boldsymbol{E}\boldsymbol{V}\dot{\boldsymbol{x}}_r(t)&= \boldsymbol{W}^T\boldsymbol{A}\boldsymbol{V} \boldsymbol{x}_r(t) + \boldsymbol{W}^T \boldsymbol{B}\boldsymbol{u}(t) \end{aligned}$$
(3a)
$$\begin{aligned} \boldsymbol{y}_r(t)&= \boldsymbol{C} \boldsymbol{V} \boldsymbol{x}_r(t), \end{aligned}$$
(3b)

the further called reduced IO-Model. The symbols \(\boldsymbol{x}_r \in \mathbb {R}^{r}\) and \(\boldsymbol{y}_r \in \mathbb {R}^{m}\) denote the reduced state and output vector.

It must be noted that the MOR techniques aim to get good approximations of the outputs \(\boldsymbol{y}\), therefore it can be stated that \(\boldsymbol{y}_r \approx \boldsymbol{y}\). Approximating the full state vector \(\boldsymbol{x}\) by \(\hat{\boldsymbol{x}}\) obtained from

$$\begin{aligned} \hat{\boldsymbol{x}} = \boldsymbol{V} \boldsymbol{x}_r \end{aligned}$$
(4)

is not necessarily accurate and should be handled with care.

Thermoelastic System Equation. Applying the FEM on the thermoelastic coupled PDE results in the following ODE-system. Note that for demonstration purposes, constant material properties are assumed.

$$\begin{aligned} \begin{bmatrix} \boldsymbol{M}_{uu} &{} 0 \\ 0 &{} 0 \end{bmatrix} \begin{bmatrix} \ddot{\boldsymbol{u}} \\ \ddot{\boldsymbol{T}} \end{bmatrix}(t) + \begin{bmatrix} \boldsymbol{D}_{uu} &{}0 \\ \boldsymbol{D}_{Tu} &{} \boldsymbol{D}_{TT} \end{bmatrix} \begin{bmatrix} \dot{\boldsymbol{u}} \\ \dot{\boldsymbol{T}} \end{bmatrix}(t) + \begin{bmatrix} \boldsymbol{K}_{uu} &{} \boldsymbol{K}_{uT} \\ 0 &{} \boldsymbol{K}_{TT} \end{bmatrix} \begin{bmatrix} \boldsymbol{u} \\ \boldsymbol{T} \end{bmatrix}(t) = \begin{bmatrix} \boldsymbol{r}_u \\ \boldsymbol{r}_T \end{bmatrix}(t) \end{aligned}$$
(5)

The vectors \(\boldsymbol{u} \in \mathbb {R}^{du}\) and \(\boldsymbol{T} \in \mathbb {R}^{dT}\) denote the deformation and temperature degrees of freedom. The matrices \(\boldsymbol{M}_{uu}, \boldsymbol{D}_{uu}, \boldsymbol{K}_{uu} \in \mathbb {R}^{du \times du}\) denote the mass, damping and stiffness matrices of the elastic system. The symbols \(\boldsymbol{D}_{TT}, \boldsymbol{K}_{TT} \in \mathbb {R}^{dT \times dT}\) are the capacity and conductivity matrices of the thermal system. The load vector \(\boldsymbol{r}\) is split into its elastic part \(\boldsymbol{r}_u \in \mathbb {R}^{du}\) and thermal part \(\boldsymbol{r}_T \in \mathbb {R}^{dT}\). Furthermore, two field coupling matrices, \(\boldsymbol{D}_{Tu} \in \mathbb {R}^{dT \times du}\) and \(\boldsymbol{K}_{uT} \in \mathbb {R}^{du \times dT}\) exist. For lots of thermoelastic problems, another simplification is possible. By assuming that the deformation damping effects generate few heat, the matrix \(\boldsymbol{D}_{Tu}\) can be neglected and both physical domains can be evaluated sequentially. First, the temperature field is calculated. Then, the resulting temperatures are used as input vector to calculate the right hand side of the deformation equation. The resulting systems of equations are

$$\begin{aligned} \boldsymbol{D}_{TT} \dot{\boldsymbol{T}} + \boldsymbol{K}_{TT}\boldsymbol{T}&= \boldsymbol{r}_T \end{aligned}$$
(6a)
$$\begin{aligned} \boldsymbol{M}_{uu} \ddot{\boldsymbol{u}}+ \boldsymbol{D}_{uu}\dot{\boldsymbol{u}} + \boldsymbol{K}_{uu}\boldsymbol{u}&= \boldsymbol{r}_u -\boldsymbol{K}_{uT}\boldsymbol{T}. \end{aligned}$$
(6b)

This approach allows the calculation of the time accurate deformation field by sequential integration of the Eqs. (6a) and (6b).

Neglecting all dynamic effects results in the equations,

$$\begin{aligned} \boldsymbol{K}_{TT}\boldsymbol{T}&= \boldsymbol{r}_T \end{aligned}$$
(7a)
$$\begin{aligned} \boldsymbol{K}_{uu}\boldsymbol{u}&= \boldsymbol{r}_u -\boldsymbol{K}_{uT}\boldsymbol{T}. \end{aligned}$$
(7b)

By calculating the temperature field at a given point in time and then applying (7b), it is possible to calculate the quasi static deformation field.

3 Modeling and Simulation Framework

After introducing the main model usages and giving a short overview of the theoretical background to simulate the time propagation of systems describable with PDEs and discretized with FEM, an overview of the tools and methods developed and used in the CRC/TR96 to model and simulate the machine tool behavior will be given next.

3.1 Concept

The modeling and simulation process typically consists of multiple steps which demand different software components. CAE-Tools like Ansys® integrate all the necessary components into one software solution to carry out most of the engineering demands but do not offer much flexibility. The various research questions and model applications of the CRC/TR96 projects demand a modular approach that allows a simple model exchange and cooperation, even between experts from different fields like mathematicians, physicists and engineers.

To solve this task, a new modeling approach using json files with custom schemes was developed. These files serve as the interfaces between different software components and allow the independent development of those by different contributors.

The most important building block is the coupled problem description file which is used to exchange system PDE-problems. It contains all information describing the model geometry like coordinate systems, FE-meshes and parts along with materials, loads, constraints, part couplings, movements etc. A detailed overview of the format, along with its application is given in [10, 14] and [15].

Fig. 1.
figure 1

Software components along with file formats as interfaces

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The different modules, along with all files that are used as interfaces are shown in Fig. 1. Each module takes the relevant model data and produces inputs for the next one. Along with all necessary model and task specific data, module dependent settings files can be provided to customize the module behavior. Since each individual module can operate and be developed independently, flexible, problem specific toolchains can be created very efficiently. The knowledge of experts can be encapsulated and made accessible for third party users.

The FE-Solver takes the coupled problem description, the output description and settings and produces the simulation result file for postprocessing tasks. The IO-Generator is used to create IO-problems according to 2 from the coupled problem description and output information. A description of the IO-format is given in [10]. MOR is performed by the MOR-Tool. It produces another form of the IO-format, additionally containing the truncation matrices. Three different solvers for the FE-Problem and the two IO-configurations produce result files which can then be processed by the postprocessing module.

Those before mentioned basic modules can be put together for more complex tasks. For the application in model predictive control scenarios, a fast model execution is needed. Therefore the simulation of the full order FE-model is not applicable. By using the IO-Generator, the MOR-Tool and a special form of the reduced model IO-Solver, it is possible to realize such a fast, controller integrable, model. A second example for the tool combination are parameter optimizations. Since a ROM has much smaller simulation times, an optimization tool incorporating IO-Generator, MOR-Tool and IO-solver can reduce the time for parameter studies significantly.

3.2 Coupled Problem Combiner

For complex systems or to speed up the development process, the possibility to split the model into smaller sub-system models is necessary. Those submodels must then be assembled efficiently to the complete system model.

The presented modeling approach using interface files allows exactly that. For each sub-system, a coupled problem description can be developed. In the next step, a combination tool takes all the sub-models and builds the assembled model description, along with a combination of the output info file.

In the CRC/TR96 such a model combiner was realized and its application could already be demonstrated for an experimental machine tool.

Fig. 2.
figure 2

Assembly process of machine tool using combination tool

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Figure 2 illustrates the described process of the model assembly. Each step needs the sub-system models along with an assembly describing json file. It contains links to the models combined, optional geometric transformations and the additional functions needed to realize the part couplings in the full model. Simulation results and comparisons of the full model and the ROM simulations for the machine shown in Fig. 2 can be found in [16].

With this approach, it is easily possible to compare models of different detail levels, reuse or swap models for different configurations.

3.3 CRC/TR96 - Realized Simulation Framework

The modeling and simulation framework developed in the CRC/TR96 to realize and demonstrate the concept described in Sect. 3.1 is shown in Fig. 3. The interoperability between open-source and proprietary tools illustrates the high flexibility of the proposed concept.

Fig. 3.
figure 3

Modeling and simulation framework of the CRC/TR96

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DUNE, an open source FE-Toolbox written in C++ [2], is used to realize the FE-solver and IO-Generator. MOR is done using a custom MOR-Tool, a collection of functions built on top of the Matrix Equation Sparse Solver (M.E.S.S.) [4]. It allows the semiautomatic calculation of reduced IO-Models from the full IO-model format. Time step integration of the IO- and reduced IO-System is done using a custom solver specifically tailored for the needs of the CRC/TR96 problems. The same mathematical structure of the IO- and reduced IO-Models results in a reduction of maintenance and implementation overhead because the same solver algorithms can be used for both model types.

Since most of the FE-modeling and -simulation realized in the CRC/TR96 is done using Ansys® Workbench (WB), a tool to make the file interface driven toolchain available for as many projects as possible was developed and shall be described next. The basis is a custom extension to WB using the Ansys® Application Customization Toolkit (ACT). By introducing new buttons and objects into the GUI of WB, it is possible to simply generate coupled problem descriptions from the modeled system. Furthermore, the definition of movements, coupling functions or integration of specialized CRC/TR96 parameterization functions is possible. After defining the model, the user has different options. Either, generate the problem description for model exchange or create a task file along with the description and call a custom module controller. This controller allows the chained call of the complete CRC/TR96 toolchain to do tasks like IO-Generation, MOR and time step simulations. The generated simulation results can be imported with another part of the ACT-Extension, allowing comparisons of Ansys® Mechanical APDL and open-source toolchain results. An example of the usage of this approach for a simple thermal model was already presented in [15]. Using the extension allowed project contributors the easy usage of the developed toolchain, even without knowledge of module internals.

4 Example Machine Tool Model

To demonstrate the developed methods and tools of the CRC/TR96 a model of the in production used machine tool DMG MORI DMU 80 eVo (DMU80eVo) was developed, which will be introduced next.

4.1 Base Model

Figure 4 shows the geometry of the machine model, along with the relevant contacts between the moving sledges. For simplification reasons the table rotation axes are fixed, resulting in a model representing a 3-axes machine tool.

Fig. 4.
figure 4

Overview of the DMU80eVo-Model with main components

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Each sledge is guided by two linear guide rails with two wagons respectively driven by a ball screw drive, resulting in 15 moving contacts. The ball screw drive spindles are supported using roller bearings, which also need special modeling efforts. Furthermore, seventeen interfaces between machine tool and fluid system exist, which for now are modeled as simple convection boundaries. Currently, the FE-model is discretized using ca. 700k elements and 1.4 million nodes, resulting in 1.4 million thermal and 4.2 million elastic degrees of freedom.

Those stats already show the high complexity of the model, resulting in a time consuming parametrization and simulation process. Especially the usage in process parallel simulations demands MOR to speed up the simulation.

4.2 Moving Contacts

Machine tools consist of multiple axis assemblies that execute large relative movements. The correct modeling of guides, drives and their internal moving contacts, providing the ability to execute those movements, is essential for a high simulation accuracy.

Linear guide wagons or ball screw drive nut assemblies and their inner parts are too complex to be modeled in a high geometrical resolution for the application in complete machine tools. The system degree of freedom would rise in a non reasonable manner. Furthermore, numerical stability issues, especially for the simulation of larger time steps, arise. The nonlinear behavior of the Hertzian deformation of the contact partners results in further overhead. The complexity of those detailed component models can be seen in publications, like [11] or [8], where the derivation of surrogate models is presented.

The common used approach for more complex systems is to replace the detailed contact geometry with a simplified, problem adapted one. The rolling parts and their respective contact behavior for thermal and elastic contacts can be replaced with surrogate models of different complexity. Ansys® Mechanical APDL allows the realization of such a surrogate model for the elastic contact by using MPC184 bushing elements. The thermal coupling can be modeled using face-face-contacts [1]. A similar approach, along with the possibility to use nonlinear coupling behavior in the contact region and showing the application of MOR techniques to speed up the simulation, is given in [17].

Using finite element couplings can significantly slow down the simulation, as contact searching is expensive and large time steps result in errors due to large relative movements per time step, which was shown and tackled using an averaging technique in [12]. By reducing the system order with MOR techniques, smaller time increments are usable, reducing the position induced errors. The application of MOR requires the definition of inputs and outputs in the zones of moving contacts, which is realized by segmenting the contact zones. The coupling model acts at the segments which are currently in contact. Figure 5 shows one such possible master-slave pair. A conflict of goals arises, as a high segment count is beneficial for the model accuracy, but as shown in [16], more segments result in a larger ROM, directly impacting the solution time per step.

Fig. 5.
figure 5

Illustration of the segmentation approach for moving contacts on the y-guide of the DMU80eVo

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Similar considerations apply for the roller bearings used in the presented model, which uses simplified models for the inner and outer bearing rings and coupling models between them.

4.3 Simulation Results

The results of a system simulation of the DMU80eVo using Ansys® Mechanical APDL are shown in Fig. 6. For confidentiality reasons, no absolute values are shown. The temperature was calculated by integrating Eq. (6a) and used as input to compute the deformations using Eq. (7b). The thermal coupling of the moving contacts was realized using face-face-contact elements [1], the elastic model uses linear MPC bushings. The friction heat in the guide is modeled by time dependent heat fluxes while heat convection to the environment is realized by using models with time dependent external temperatures and heat transfer coefficients. Heat fluxes in the table center and on the tool tip model the process heat. As can be seen, the highest temperatures result at the tool tip. The impact of the guide and drive friction is small. Further detailed discussions of the modeling approaches and results are omitted, since the optimization and validation of the model using measurements is still in progress. The application of MOR techniques with a quantitative comparison of the optimized, full order model with the ROM using the presented toolchain along with detailed modeling aspects and simulation results will be presented in a future publication.

Fig. 6.
figure 6

Calculated fields after 10000 s operation a) temperature, b) deformation

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5 Conclusion and Outlook

This contribution introduced a framework to model complex technical systems using an interface file based, modular modeling approach, allowing simple model exchange and independent development of different, task specific tools.

MOR and its usage as an important tool to increase the simulation efficiency preserving a given model accuracy were shown.

After that, a toolchain realizing the described modular approach, which was developed and used in the CRC/TR96, was presented. References to multiple usage examples, showing applications and proving the flexibility of the approach, were given.

Finally, a model and simulation results of the thermoelastic behavior of an in production machine tool, which is one of the main demonstration objects of the tools and methods developed in the CRC/TR96, were shown. Different coupling approaches, especially for moving contacts of linear guides and ball screw drives, were discussed.

Further development steps to facilitate the process parallel simulation of production processes using the presented machine tool model will be executed. This includes model validation with measurements and parameter optimization, integrating a sophisticated model of the fluid system and applying MOR techniques to speed up the simulations. Furthermore, different coupling techniques, like the one given in [17], shall be implemented and compared with the here presented approach to further enhance the model accuracy.