Efficiency of dynamic relaxation methods in form-finding of tensile membrane structures
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Abstract
The main idea of this study is to investigate and compare the efficiency of the dynamic relaxation methods (DRMs) in the form-finding of tensile membrane structures (TMSs). The form-finding process as a main stage in the design and construction of TMSs has been considered for finding an equilibrium configuration subjected to a specific prestress distribution. DRMs as a pseudo-dynamic analysis are an explicit iterative technique for the nonlinear analysis of structures. In the techniques, the static equilibrium of structures is solved by integrating the pseudo-dynamic equations in order to obtain the steady state of the pseudo-dynamic problem. In this study, the efficiency and generality performance of DRMs are compared through solving three selected TMSs. In order to achieve this purpose, seven schemes of the DR approach are selected based on combining the fictitious parameters including the time step, diagonal mass, and damping matrices which were proposed in the previous studies. Furthermore, a reference index is proposed by combining the total number of iterations and the overall duration of analysis in order to appropriately compare the schemes of the DR approach in the form-finding of TMSs.
Keywords
Tensile membrane structures Dynamic relaxation method Form-finding Nonlinear analysis Reference index1 Introduction
Tensile membrane structures (TMSs) are well known as lightweight and cost-effective structures that are used for long span roofing components, such as buildings, stadiums, and exhibition halls. Along with structural stability, TMS(s) attain an aesthetic architectural form [1, 2]. The coated fabrics utilized in the construction of a TMS are unable to resist flexure and shear forces. However, these materials can reduce not only temperature but also the costs of maintenance due to their self-cleaning properties. Such structures are able to tolerate external events through their membrane prestress in plane and anticlastic surface curvature. TMSs in comparison with traditional roofing materials also bear a lower structural load and more earthquake resistance produce due to the membrane’s elasticity. The stages of form-finding, static analysis, and patterning are considered as the preliminary design procedure of a TMS [3]. Furthermore, the initial design of a TMS includes of two primary steps: (1) determining an equilibrium form of a TMS and (2) implementing analysis of the form subjected to its service condition loads [4].
In order to implement the form-finding process, several numerical methods have been proposed, such as the force density (FD) method [5], dynamic relaxation method (DRM) [6, 7], updated reference strategy [8] and particle spring systems [9]. At first, these methods were adopted and developed for implementing the form-finding of prestressed membrane structures. In recent years, significant methods have been developed and several improvements have been proposed. To find the equilibrium form of membrane structures, Sheck [10] introduced a simple linear method based on force densities. In this method, it was assumed that a structure can be modeled and considered by cable networks. Employing the surface stress density method as an iterative procedure, Maurin and Motro [11] proposed a new form-finding method for TMSs. In the surface stress density method, an isotropic stress tensor and an iterative procedure were considered for triangular elements. By achieving the convergence of the procedure, configurations were concluded that satisfy the conditions of the static equilibrium laws. Sanchez et al. [12] presented a novel approach for the conceptual design of fabric tensile structures. In order to obtain preliminary shapes of tensile structures, the proposed approach combined a form-finding method with a surface fitting approach. With the FD method, the value of force density often depends on the researcher’s experience and can only be determined after several trial calculations. Hence, Ye et al. [13] introduced a modified FD approach for the form-finding of membrane structures. According to their method, membrane stress and cable tension were utilized as initial conditions instead of as the assumed value of FD approach (i.e., the quantitative relationships between membrane stress, cable tension, and force density were established), and the unbalanced force of each node was employed to control the error. Barnes [14] described a DRM-based numerical procedure with kinetic damping (called the kinetic DRM) for the form-finding, analysis, and fabrication patterning of TMDs. In the kinetic DRM, damping factor is equal to zero. Passing from a peak point of the kinetic energy diagram indicates that the kinetic energy has a reducing trend. Using the result of this time leads to divergence process. From the viewpoint of a practicing membrane engineer, Wakefield [7] reviewed the selection and application of appropriate analysis techniques and demonstrated that DRM may be considered as the preferred form-finding method. Lewis [15] discussed the advantages and limitations of three approaches, namely transient stiffness, FD, and DRM. In addition, the study provided insight into their applicability as the numerical tools of design in fabric structures. Recently, Alic and Persson [16] introduced a form-finding method established by a hybrid of DRM and isogeometric membrane elements. The elements were developed based on non-uniform rational B-splines (NURBS). The results of this study revealed that the discretization and shape of elements effectively impressed on the form-finding. This procedure can be used to quickly find form-finding since, in NURBS’ description of the curved geometry well, form-finding may be performed with a coarse mesh. Other engineering problems have also demonstrated the successful application of isogeometric analysis [17, 18].
The main contribution of the present study is to investigate the effectiveness of different DRM-based strategies in the form-finding of TMSs. In the DRM techniques, the fictitious mass, damping, and time step factors impress on their stability and speed of convergence. Hence, the efficiency and generality performance of DRMs based on combining the proposed fictitious parameters including the time step, diagonal mass, and damping matrices are compared through solving three selected TMSs. Furthermore, a reference index as an efficient criterion is proposed by the combination of the total number of iterations and overall analysis duration. The results show that the reference index can be adopted for comparing the generality performance of DRMs. In addition, the DR approaches can be efficiently used as a suitable tool in the form-finding of TMSs.
2 Standard dynamic relaxation approach
- (1)
Defining \(\varepsilon ,\) \({\mathbf{X}}^{0} ,\) \({\dot{\mathbf{X}}}^{ - 1/2} = {\mathbf{0}}\) and \(\,t^{0} .\)
- (2)$$n = 0.$$
- (3)
Calculating the internal force vector and applying boundary conditions.
- (4)
Determining the residual forces, artificial mass and damping matrices.
- (5)
Updating the time step t.
- (6)
Calculating \({\dot{\mathbf{X}}}^{n + 1/2}\) and \({\mathbf{X}}^{n + 1} .\)
- (7)
If \(\left\| {{\mathbf{R}}^{n + 1} } \right\| \le \varepsilon ,\) stop the DR algorithm.
- (8)$$n = n + 1.$$
- (9)
If \(n \le n_{\hbox{max} } ,\) continue the iteration of the DRM from the step (3).
3 Modifications of the standard DRM
The iterations are inherently unstable in the standard DRM because the numerical time integration is utilized for solving the differential equations of motion [23, 24, 25, 26]. Hence, the stability of DRM depends on fictitious parameters including the time step, diagonal mass, and damping matrices. Therefore, new techniques have been proposed for calculating these fictitious parameters. In the following section, some of the selected techniques are expressed.
3.1 Optimal time step
It should be noted that the second-order derivative of Z_{R} with respect to \(t^{n + 1}\) is always positive so that Eq. (11) produces the highest convergence rate. Sometimes, an extremely small or a very large value is obtained for the time step, which may cause numerical instability [25].
3.2 The modified DRM
In fact, the mdDR technique can enhance the convergence rate of DRM and can reduce its computational cost [23].
3.3 Fictitious damping factor technique
It should be noted that the lowest eigenvalue is first obtained from Eq. (16) and is compared with that of Rayleigh concept. Finally, the minimum value is utilized for updating the artificial damping factor.
3.4 Residual energy minimizer time step
Thus, one of two values of Eq. (19) is selected based on the satisfaction of the condition which is introduced in Eq. (23).
4 Finite element method for the nonlinear analysis of TMS
This section presents the finite element method for the nonlinear analysis of TMSs. For this purpose, the plane-stress triangular element is employed for describing three-dimensional membranes. Furthermore, the approach for obtaining the geometric stiffness matrix of membrane structures is presented.
4.1 The plane-stress triangular element
4.2 The geometric stiffness matrix of plane-stress triangular element
In fact, the explicit elastic and geometric stiffness matrices are provided by using Eqs. (25) and (30) for plane elasticity problems. In the next stage, three-dimensional rotational effects of each element should be considered. This issue is expressed in the following section presents.
4.3 Geometric stiffness matrix in 3-D space
The more detail of this approach can be found in Spillers et al. [30].
5 Form-finding procedure of TMS
Schemes of the DR technique
Scheme | Fictitious parameters | ||
---|---|---|---|
Mass | Damping | Time | |
DRM 1 | Equation (6) | Equation (7) | t^{k}= 1 |
DRM 2 | Equation (6) | Equation (7) | Equation (11) |
DRM 3 | Equation (6) | Equation (7) | Equation (19) |
DRM 4 | Equation (13) | Equation (16) | t^{k}= 1 |
DRM 5 | Equation (13) | Equation (7) | t^{k}= 1 |
DRM 6 | Equation (13) | Equation (14) | t^{k}= 1 |
DRM 7 | Equation (13) | Equation (16) | Equation (19) |
In fact, the criterion can be efficiently used as an appropriate and convenient criterion instead of two criteria. A higher value of the RI parameter represents the scheme with the highest performance.
6 Illustrative examples
The schemes of DRM shown in Table 1 were applied to three different examples. All DR schemes were programmed in FORTRAN and calculations were carried out on the computer with processor AMDE-450 1.65 GHz, 4 GB RAM. The acceptable residual error (i.e. ε) was same for all solutions and was assumed to be equal to 10^{−4}. In all examples, the structure was discretized using triangular elements, and the surface topology was defined by a trial shape which is used in the first step of the form-finding analysis .
6.1 Spherical cap
Comparing the results of the DR schemes for the loading condition Case 1
Scheme | Number of iterations in each loading step loading | \(I^{i}\) | \(E_{I}^{i}\) | \(T^{i}\)(s) | \(E_{T}^{i}\) | RI | |||
---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | ||||||
DRM 1 | 63 | 57 | 55 | 54 | 229 | 0 | 1.41 | 22.97 | 11.48 |
DRM 2 | 33 | 27 | 28 | 31 | 119 | 100 | 1.04 | 72.97 | 86.48 |
DRM 3 | 31 | 33 | 31 | 36 | 131 | 89.09 | 0.84 | 100 | 94.55 |
DRM 4 | 38 | 35 | 36 | 38 | 147 | 74.54 | 1.26 | 43.24 | 58.89 |
DRM 5 | 60 | 55 | 53 | 52 | 220 | 8.18 | 1.58 | 0 | 4.09 |
DRM 6 | 54 | 50 | 48 | 47 | 199 | 27.27 | 0.92 | 89.18 | 58.22 |
DRM 7 | 30 | 36 | 42 | 39 | 147 | 74.54 | 0.84 | 100 | 87.27 |
Comparing the results of the DR schemes for the loading condition Case 2
Scheme | Number of iterations in each loading step loading | \(I^{i}\) | \(E_{I}^{i}\) | \(T^{i}\)(s) | \(E_{T}^{i}\) | RI | |||
---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | ||||||
DRM 1 | 41 | 39 | 39 | 41 | 160 | 6.25 | 1.04 | 38.63 | 22.44 |
DRM 2 | 37 | 39 | 38 | 38 | 152 | 56.25 | 1.05 | 36.36 | 46.310 |
DRM 3 | 39 | 38 | 40 | 41 | 158 | 18.75 | 1.21 | 0 | 9.38 |
DRM 4 | 41 | 38 | 41 | 41 | 161 | 0 | 0.96 | 56.81 | 28.41 |
DRM 5 | 36 | 35 | 36 | 40 | 145 | 100 | 0.87 | 77.27 | 88.63 |
DRM 6 | 37 | 38 | 38 | 38 | 151 | 62.50 | 0.77 | 100 | 81.25 |
DRM 7 | 38 | 36 | 38 | 38 | 150 | 68.75 | 1.00 | 47.72 | 58.23 |
6.2 A fat stretched membrane
Comparing the results of the DR schemes for the loading condition Case 1
Scheme | Number of iterations in each loading step loading | \(I^{i}\) | \(E_{I}^{i}\) | \(T^{i}\)(s) | \(E_{T}^{i}\) | RI | |||
---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | ||||||
DRM 1 | 68 | 70 | 72 | 73 | 283 | 64 | 1.68 | 41.57 | 52.79 |
DRM 2 | 70 | 59 | 69 | 70 | 262 | 92 | 1.44 | 68.54 | 80.27 |
DRM 3 | 78 | 70 | 93 | 84 | 325 | 8 | 2.05 | 0 | 4 |
DRM 4 | 70 | 70 | 71 | 71 | 282 | 65.33 | 1.77 | 31.46 | 48.39 |
DRM 5 | 71 | 65 | 57 | 65 | 283 | 64 | 1.16 | 100 | 82 |
DRM 6 | 69 | 59 | 63 | 65 | 256 | 100 | 1.41 | 71.91 | 85.96 |
DRM 7 | 82 | 86 | 74 | 89 | 331 | 0 | 1.88 | 19.10 | 9.55 |
Comparing the results of the DR schemes for the loading condition Case 2
Scheme | Number of iterations in each loading step loading | \(I^{i}\) | \(E_{I}^{i}\) | \(T^{i}\)(s) | \(E_{T}^{i}\) | RI | |||
---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | ||||||
DRM 1 | 68 | 70 | 72 | 73 | 283 | 64 | 1.9 | 20.95 | 42.48 |
DRM 2 | 70 | 59 | 62 | 70 | 262 | 92 | 1.44 | 64.76 | 78.38 |
DRM 3 | 78 | 70 | 93 | 84 | 325 | 8 | 2.12 | 0 | 4 |
DRM 4 | 70 | 70 | 71 | 71 | 282 | 65.33 | 1.55 | 54.28 | 59.81 |
DRM 5 | 71 | 65 | 57 | 65 | 258 | 97.33 | 1.2 | 87.62 | 92.48 |
DRM 6 | 69 | 59 | 63 | 65 | 256 | 100 | 1.07 | 100 | 100 |
DRM 7 | 82 | 86 | 74 | 89 | 331 | 0 | 1.91 | 20.00 | 10 |
6.3 Practical example: Kresge auditorium
These are the fixed nodes which were used in the DR schemes. While the plan dimensions and fixed elevations were based on the Kresge auditorium, the prescribed load and other parameters were chosen arbitrarily to generate a shape just for the purposes of visualization. The Young’s modulus was considered as 10,000 ksi while the Poisson’s ratio was 0.3. The thickness of this structure was taken as 0.01576 in. The initial prestress values in the warp and fill directions was assumed equal to 300 ksi. The load was applied at inner joints, which was equal to 10 kip.
Comparing the results of the DR schemes for the form-finding of the Kresge auditorium
Scheme | Number of iterations in each loading step loading | \(I^{i}\) | \(E_{I}^{i}\) | \(T^{i}\)(s) | \(E_{T}^{i}\) | RI | |||
---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | ||||||
DRM 1 | 126 | 127 | 137 | 173 | 563 | 0 | 1.92 | 39.53 | 19.77 |
DRM 2 | 58 | 89 | 81 | 100 | 328 | 74.84 | 1.56 | 67.44 | 71.14 |
DRM 3 | 49 | 83 | 89 | 101 | 322 | 76.75 | 1.24 | 92.25 | 84.49 |
DRM 4 | 49 | 54 | 60 | 81 | 249 | 100 | 1.14 | 100 | 100 |
DRM 5 | 118 | 121 | 130 | 165 | 534 | 9.24 | 1.85 | 44.96 | 27.09 |
DRM 6 | 106 | 113 | 124 | 160 | 503 | 19.11 | 2.35 | 6.21 | 12.65 |
DRM 7 | 72 | 98 | 150 | 215 | 535 | 8.98 | 2.43 | 0 | 4.46 |
7 Discussion
Value of the RI average for the DR sachems
Scheme | Average of RI | G_{i} |
---|---|---|
DRM 1 | 29.792 | 7 |
DRM 2 | 72.516 | 1 |
DRM 3 | 39.284 | 5 |
DRM 4 | 59.1 | 3 |
DRM 5 | 58.858 | 4 |
DRM 6 | 67.616 | 2 |
DRM 7 | 33.902 | 6 |
Column G_{i} lists the grades of each scheme based on the RI average. Based on the value of the RI average, the schemes are graded from one to seven, as indicated in Table 7. According to the obtained grades, the higher ranking schemes with the good performance are schemes 2, 6, 4, and 3, respectively. Furthermore, scheme 1 (i.e., standard DRM) shows the worst performance.
8 Conclusions
The results confirm that comparing both the total number of iterations and the overall duration of analysis is not appropriate for the purposes of the study. Hence, the proposed RI criterion instead of two criteria is adopted as the best criterion for assessing the efficiency of different DRMs.
In the more cases of the study’s examples, the results imply that the efficiency of the standard DRM (scheme 1) is not suitable in comparison with other schemes. Thus, the standard DRM can not be considered as a reliable approach in the TMS form-finding.
Based on the obtained average of the RI, the schemes 2, 6, 4, and 3 have the high performance and can be recognized as the reliable schemes in the form-finding of TMSs.
It is worth emphasizing that the best performance of the form-finding is obtained by the scheme 2 (i.e., the optimal time step approch proposed by Kadkhodayan et al. [27].
Notes
Compliance with ethical standards
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
References
- 1.Lewis WJ (2003) Tension structures form and behavior. Thomas Telford, LondonCrossRefGoogle Scholar
- 2.Dutta S, Ghosh S, Inamdar MM (2017) Reliability-based design optimization of frame-supported tensile membrane structures. J Risk Uncertainty Eng Syst Part A Civ Eng 3(2):1–11Google Scholar
- 3.Lewis WJ (2013) Modeling of fabric structures and associated design issues. J Archit Eng ASCE 19(2):81–88CrossRefGoogle Scholar
- 4.Huntington CG (2013) Tensile fabric structures: design, analysis and construction. American Society of Civil Engineers, RestonCrossRefGoogle Scholar
- 5.Linkwitz K (1999) Form finding by the direct approach and pertinent strategies for the conceptual design of pre-stressed and hanging structures. Int J Space Struct 14(2):73–87CrossRefGoogle Scholar
- 6.Barnes MR (1994) Form and stress engineering of tension structures. Struct Eng Rev 6(3-4):175–202Google Scholar
- 7.Wakefield D (1999) Engineering analysis of tension structures: theory and practice. Eng Struct 21(8):680–690CrossRefGoogle Scholar
- 8.Bletzinger KU, Ramm E (1999) A general finite element approach to the form finding of tensile structures by the updated reference strategy. Int J Space Struct 14(2):131–145CrossRefGoogle Scholar
- 9.Kilian A, Ochsendorf J (2005) Particle-spring systems for structural form finding. J Int Assoc Shell Spatial Struct 46(148):77–84Google Scholar
- 10.Schek HJ (1974) The force density method for form finding and computations of general networks. Comput Methods Appl Mech Eng 3:115–134MathSciNetCrossRefGoogle Scholar
- 11.Maurin B, Motro R (1998) The surface stress density method as a form-finding tool for tensile membranes. Eng Struct 20(8):712–719CrossRefGoogle Scholar
- 12.Sanchez J, Serna MA, Morer P (2007) A multi-step force-density method and surface fitting approach for the preliminary shape design of tensile structures. Eng Struct 8:1966–1976CrossRefGoogle Scholar
- 13.Ye J, Feng R, Zhou S, Tian J (2012) The modified force-density method for form-finding of membrane structures. Int J Steel Struct 12(3):299–310CrossRefGoogle Scholar
- 14.Barnes MR (1999) Form finding and analysis of tension structures by dynamic relaxation. Int J Space Struct 14(2):89–104CrossRefGoogle Scholar
- 15.Lewis WJ (2008) Computational form-finding methods for fabric structures. Eng Comput Mech 161(3):139–149Google Scholar
- 16.Alic V, Persson K (2016) Form finding with dynamic relaxation and isogeometric membrane elements. Comput Methods Appl Mech Eng 300:734–747MathSciNetCrossRefGoogle Scholar
- 17.Roodsarabi M, Khatibinia M, Sarafrazi SR (2016) Hybrid of topological derivative-based level set method and isogeometric analysis for structural topology optimization. Steel Compos Struct 21(6):1287–1306CrossRefGoogle Scholar
- 18.Khatibinia M, Roudsarabi M, Barati M (2018) Topology optimization of plane structures using binary level set method and isogeometric analysis. Int J Opt Civ Eng 8(2):209–226Google Scholar
- 19.Day AS (1965) An introduction to dynamic relaxation. The Engineer 219:218–221Google Scholar
- 20.Topping BHV, Ivanyi P (2007) Computer aided design of cable membrane structures. Saxe-Coburg Publications, KippenGoogle Scholar
- 21.Rezaiee-Pajand M, Sarafrazi SR (2010) Nonlinear structural analysis using dynamic relaxation method with improved convergence rate. Int J Numer Methods Eng 7:627–654MathSciNetzbMATHGoogle Scholar
- 22.Zhang LC, Yu TX (1989) Modified adaptive dynamic relaxation method and its application to elasticplastic bending and wrinkling of circular plates. Comput Struct 34:609–614CrossRefGoogle Scholar
- 23.Rezaiee-Pajand M, Alamatian J (2008) Nonlinear dynamic analysis by dynamic relaxation method. Struct Eng Mech 28:549–570CrossRefGoogle Scholar
- 24.Rezaiee-Pajand M, Sarafrazi SR, Rezaiee H (2012) Efficiency of dynamic relaxation methods in nonlinear analysis of truss and frame structures. Comput Struct 112-113:295–310CrossRefGoogle Scholar
- 25.Rezaiee-Pajand M, Kadkhodayan M, Alamatian J (2012) Timestep selection for dynamic relaxation method. Mech Based Des Struc 40(1):42–72CrossRefGoogle Scholar
- 26.Alamatian J, Hosseini-Nejad Goshik M (2016) An effcient explicit framework for determining the lowest structural buckling load using dynamic relaxation method. Mech Based Des Struc 45(4):1–12Google Scholar
- 27.Kadkhodayan M, Alamatian J, Turvey GJ (2008) A new fictitious time for the dynamic relaxation (DXDR) method. Int J Numer Methods Eng 74:996–1018CrossRefGoogle Scholar
- 28.Sarafrazi SR (2012) Numerical integration for structural dynamic analysis. PhD Dissertation, Department of Civil Engineering, Ferdowsi University, Iran. (in Persian)Google Scholar
- 29.Zienkiewicz OC (1977) The finite element method. McGraw-Hill, New JerseyzbMATHGoogle Scholar
- 30.Spillers WR, Schlogel M, Pilla D (1992) A simple membrane finite element. Comput Methods Appl Mech Eng 45(1):181–183Google Scholar
- 31.