software program for continuous optimization process. ... An advanced technological aid for the designer and development engineer is the structural optimization ...
Structural Optimization of Solid Components with Topology Optimizing Structural Language TopoSLang Dirk Roos, Johannes Will CAD-FEM GmbH, Grafing, Germany Frank Vogel inuTech GmbH, Seukendorf, Germany
Summary In the design process the basic structural layout can be found using the structural optimization, such as topology and shape optimization. During the last years numerous methods and software packages have been developed. However, since these tools are commonly only able to determine the rough form of the body, the results of topology optimization have to be further processed and translated by hand or by additional programs. Therefore, the objective of the development of the presented software program TopoSLang (Topology optimizing Structural Language) is to introduce an easy to use software program for continuous optimization process. The proposed optimization methods provide an automatic link from topology and shape optimization up to the geometry description in the frame of existing CAD software. TopoSLang supports sophisticated topological optimization, mesh smoothing, shape optimization in respect to stress relaxation and the automatic generation of STL-geometry description to transfer the optimization results back to CAD. The user-friendly preparation of mathematical algorithms improves the efficiency and extends the applicability of the conventional optimization procedures.
Keywords Structural optimization, topology optimization, shape optimization, mesh smoothing, biological growth, software development
19th CAD-FEM Users’ Meeting 2001 International Congress on FEM Technology
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October 17-19, 2001 Hotel Dorint Sanssouci Berlin, Potsdam
0.
Introduction
The determination of the structural layout is the basic problem of the design process. The numerical simulation based on the finite element method can be utilized in the process of computer-aided design. An advanced technological aid for the designer and development engineer is the structural optimization, which creates weight-optimized design. However the existing finite element and optimization technologies still have limitations to be practically used in the design process. The main problem is a user-friendly continuous workflow from topology optimization back to a CAD-model. This is the motivation of the cooperation project “Development of a user-friendly finite element tool for design optimization of solid components (funded by the Arbeitsgemeinschaft industrieller Forschung AIF project ProINNO)”. The involved parties are the CAD-FEM GmbH, the inuTech GmbH, the Mathematical Institute of the University of Bayreuth and the Institute of Structural Mechanics of the Bauhaus University Weimar. The objective of the project is to advance topology and shape optimization strategies for their user-friendly use in the early design process. All software developments will be included in the software architecture of SLang (Structural Language) and combined with user-friendly graphical user interface. Result of the software development is a software platform for topology optimization, called TopoSLang.
1.
Topology Optimization
In topology optimization the structural geometry is described by "0 - 1" material distribution in a given design space. In finite element analysis the material distribution can be interpreted as the element pseudo-density distribution. The goal of the topology optimization is to find the pseudo-density distribution, which minimizes the strain energy (optimal compliance design) or respectively results in an optimized stiffness (as shown in Fig. 1 and 2). In [6] a mathematically method is developed to solve these kinds of ill-posed non-convex material distribution optimization problems by combining three disciplines, the structural optimization, the relaxation of non-convex functional and homogenization of micro-structured materials. The material is replaced by an equivalent homogenous substitute. The microstructure is to be assumed a square (cube) with a square (cube) hole built of isotropic material. By changing the hole size of the microstructure, volume variations between zero and one are possible. In [2] the homogenization method is used to obtain the material properties of the substitute. However, the energy approach by [9] is employed in this development. This energy approach is based on the assumption that the strain energy of the square (cube) micro cell and its filled substitute are identical for any state of local strain. Thus we have to apply the standard local states of strain as prescribed boundary deformations, and to compute the strain energy values for the micro cell domain by applying finite element analysis at this microscopic level (see [9] for details).
Fig. 1: The given design space could be a plane structure subjected to a vertical load V.
19th CAD-FEM Users’ Meeting 2001 International Congress on FEM Technology
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Fig. 2: The topology optimization gives the "0 - 1" material distribution interpreted as the element pseudo-density distribution. In order to optimize the compliance subjected to a volume constraint, an optimality criterion (OC) approach can be used [10]. This iterative method is based on the necessary conditions of first order, which can be derived by means of first variation of an augmented Lagrangian of the underlying continuous compliance optimization problem. The handling of multiple loading and the algorithmic code conversion is specified in [10, 8]. The developed OC approach is very fast and efficient - largescale optimization problems with more than 100.000 elements can be solved. A filter strategy (member size control) is implemented which removes the checker boarding effect using finite elements with linear shape functions and controls the solidity of the resulting structure. With this member size control the objectionable influence of the mesh density to the results is strongly relaxed. For some problems in topological optimization consideration of stress constraints would be very helpful. Unfortunately, using OC only one constraint can be considered in the optimization problem. To overcome this limitation we implemented a sequential convex programming (SCP) approach [12]. In every optimization iteration with SCP, a local convex approximation is considered and solved. The SCP approach is very flexible since theoretically any arbitrary topology optimization can be solved: multiple constraints, e.g. element stress constraints, any kind of objective, e.g. frequency optimization. Again a filter strategy (member size control) is implemented to avoid checker boarding and control solidity, as shown in Fig. 3.
Fig. 3: Material distribution in case of linear element shape functions without filter and using filter approach. With sophisticated mathematical algorithms it is possible to use linear elements for large-scale problems, to avoid objectionable mesh dependence, to control solidity and to consider stress constraints.
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2.
Mesh smoothing
The resulting surfaces after successful topological optimization, as shown in Fig. 4 are normally rough and has to be smoothed. In TopoSLang an automatic smoothing procedure is available. All points of the outer surface will be moved on a local least square second order polynomial The polynomial approximates points located in the near of the considered point. A user-defined radius selects the points. The local least square second order polynomial is oriented on a least square plain with u,v,wcoordinates. In order to avoid any degeneration of the finite elements, the mesh in the near of the viewed point will be modified using static condensation. Fig. 5 shows the smoothed finite element mesh.
Fig. 5: The finite element mesh after mesh smoothing.
Fig. 4: The finite element mesh after topology optimization.
After smoothing the optimized structure and FE-mesh relaxation analysis to check stress states can be performed.
3.
Shape optimization in respect to stress relaxation using biological growth
After smoothing there may be still stresses exceeding critical stress states. For relaxation of these stress peaks we use biological growth algorithms. The natural evaluation leads towards a uniform stress state. That implies that no part of a structure is overstressed or under stressed. Living biological structures adaptively achieve a nearconstant or uniform stress state. This is important because cracks and fatigue usually occur in regions of high stress. Fig. 6 and 7 show an example of a biological growth optimization problem with 17 % stress reduction.
Fig. 6: The Von Mises stress distribution of the Fig. 7: The resulting structure and the Von Mises initial structure. stress distribution after the biological growth optimization.
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Using volumetric swelling of regions with concentrated stress state, we obtain structures with increased lifetime in case of frequent cyclic loadings. On the other hand, using volumetric shrinking of low stress regions, no structural material is wasted. Mattheck [7] developed a biological growth method by volumetrically swelling or shrinking of a structure according to its stress distribution. This computational method can be easily realized using pseudo-thermal strains Thus, two finite element analysis runs are required per optimization step. The first step is the swelling/shrinking iteration step using the actual pseudo-thermal strains. For the swelling/shrinking iteration step we can use a separate set of restraint conditions. The second iteration step is based on the updated nodal coordinates with respect to the displacements of the swelling/shrinking analysis.
4.
Transfer of new geometry via STL-format
A very important part of a continuous topology optimization is the transfer of the new geometry approximation back to the CAD-environment. After testing of commercially available Reverse Engineering tools we did not found any procedure to extract higher order CAD surfaces like B-Spline surfaces with an acceptable amount of user interaction. Because of that, we decided to extract the smoothed surface approximation of the optimized structure as STL-format. This standard format can easily be extracted fully automatically and imported to any CAD-program. The TopoSLang STLsurfaces create watertight bodies, which can be processed as volumes in CAD-programs (see Fig.21).
5.
Software architecture based on SLang – the Structural Language
The software for structural optimization analysis should encompass state-of-the-art technology in both the structural FE-analysis as well as optimization analysis parts. This does not necessarily imply that one single program package must include everything. Because we wanted to have the possibility of connecting arbitrary external FE-solvers, Preprocessors or CAD-programs we choose to develop an optimization platform. Additionally it seems useful to formulate software tasks in small, easy to control steps. It should be possible to combine these steps into larger segments, which can be executed repeatedly. Such a software solution can be achieved by implementing a problem-oriented module set in which the individual modules pertain to optimization and structural analysis. A dedicated modular software package for stochastic structural analysis and optimization (SLang) along this line has conceptually been presented several years ago by [3] and substantiated subsequently by [4]. In a way, SLang can be seen as a toolbox containing the basic software products for both optimization and finite element analysis, which can interact smoothly and transparently. SLang integrates finite elements and optimization at a level, which appears to be sufficient for a wide range of engineering problems. In addition, the recent developments in SLang allow further enhancement regarding non-linear optimization [13], interprocess communication and parallel processing, which can be used advantageously for structural optimization analysis. In order to meet the above-mentioned requirements regarding interaction between optimization and finite element analysis, SLang [1] has been chosen as basic software architecture for the development of the optimization platform TopoSLang. The methods of topology optimization, biological growth and mesh smoothing are included in the SLang software and a user-friendly GUI was developed. The resulting software sub-package of SLang is called TopoSLang. TopoSLang also includes the SLang FE-solver. Because of that, after preparing the problem (meshing, loading, constrains) optionally all analysis can be done in TopoSLang or external solver can be used. Presently the ANSYS solver is connected.
6.
Interfacing with other finite element software packages
Some user may wish to use arbitrary efficient and fast finite element solvers. For that reason we defined an application programming interface (API) and so far connected the ANSYS solver to TopoSLang. Analytical gradient information is needed for the OC and SCPIP approach. We can connect external solvers in two ways. In case of compute the element gradients in ANSYS we use ANSYS user programmable features (UPF). To avoid programming of external solvers we can also use the internal finite element structure of TopoSLang. In case of a complete reproduction of external FE-models the element gradients can be calculated without any modifications of external solvers. Then the interfacing is realized using data files, as shown in Fig. 8.
19th CAD-FEM Users’ Meeting 2001 International Congress on FEM Technology
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ANSYS
Commands & ANSYS Parametric Design Language
data files Commands & SLang Macro Programming
SLang the Structural Language
Fig. 8: Interfacing between TopoSLang and ANSYS using APDL/MACRO programming. For import of the finite element model a standardized interface between ANSYS and TopoSLang using ANSYS ASCII-db format (*.cdb) was developed. An example of FE transfer can be seen in Fig. 9.
Fig. 9: Finite element model exchange from ANSYS to TopoSLang.
19th CAD-FEM Users’ Meeting 2001 International Congress on FEM Technology
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7.
Graphical User Interface
A system independent customer-friendly user interface has been developed to facilitate the use of the optimization platform. In Tab. 1 the considered alternatives of GUI libraries are listed.
Library
NT
Unix
Linux
Mac
Windows API
+
-
-
-
GTK
-
+
+
+
Motif
+
+
+
-
Mac API
-
-
-
+
JavaSwing/3D
+
+
+
+
wxWindows
+
+
+
+
Qt
+
+
+
+
Tab. 1: GUI libraries The highest possible system independence was our dominant criterion. To avoid parallel development effort for different hardware platforms the use of Windows API, GTK, Motif and Mac API was ruled out.
Fig. 10: Qt based GUI. 19th CAD-FEM Users’ Meeting 2001 International Congress on FEM Technology
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October 17-19, 2001 Hotel Dorint Sanssouci Berlin, Potsdam
Java Swing in connection with Java3D, widely used in current software developments has been tested, though its performance has been judged unsatisfactory for large-scale problems. With the GUI library wxWindows during the GUI development-compiling errors occurred frequently. Thus we decided to use Qt. Qt is a popular commercial GUI toolkit. Additionally, it supports client-server programming via TCP/IP sockets. Thereby, a job distribution even in heterogeneous networks is possible. The Qt based GUI is currently developed to provide topology optimization, biological growth and mesh smoothing. Fig. 10 shows a typical QT based GUI of the optimization platform. By this way, the user will be continuously supported in optimizing solid components, so profound mathematical knowledge is not necessary. The user focus on definition of the physical part of the optimization problem, i.e. objectives and constrains definition. For the mathematical parameters of the optimization proper default values are defined. Highest priority in the interface design has the user-frienly support of the topology optimization workflow. The user will be continuously supported in optimizing solid components, so profound mathematical knowledge is not necessary. The user can focus on the definition of the physical part of the optimization problem, i.e. weigth reduction or load case combination. For all mathematical parameters of the optimization algorithms proper default values are defined.
8.
An illustrating example and workflow of the optimization process
8.1
Preparing the problem
TopoSLang assumes that the user meshes the 3D-geometry with solid or tetrahedral elements, defines the physical properties (Young’s Modulus, weight, Poisson’s ratio), the loading (different load cases) and the constraints in an external preprocessor or CAD-program. At the moment we support the ANSYS preprocessor to define the optimization problem. Because only a translation of ASCII-file information about FE-discrimination, material constants, constrains and loads have to be performed we can connect easily other preprocessors or CAD-programs. 8.2
Workflow from topology optimization, mesh smoothing and shape optimization up to the CAD export
The following simple structure subjected to loads as shown in Fig. 11 is to be optimized using topology and shape optimization. The user can define the areas that shall be subject to topology optimization and areas that – for any constructive reasons – should not be modified. In Fig. 12, the modifiable areas are colored green (lighter in black and with print).
Fig. 11: Example structure with load and support conditions.
Fig. 12: Definition of modifiable (green) and unchangeable (red) areas.
In the first step, the structure will now be subjected to topology optimization. In this case, the user has defined the degree of material saving as 50%, stress and frequency constrains. The topology dialog is shown in Fig. 13.
19th CAD-FEM Users’ Meeting 2001 International Congress on FEM Technology
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Fig. 13 Topology optimization is simple defined by objective function, constrains and default optimization parameters. Fig. 14 shows the pseudo-density distribution calculated considering the given restraints. The resulting non-smooth finite element mesh is displayed in Fig. 15.
Fig. 14: Pseudo-density distribution.
Fig. 15: Resulting finite element mesh.
After this, the finite element mesh is smoothed. Here, the user may define the surfaces to be smoothed and the radius of smoothing (see Fig. 16).
19th CAD-FEM Users’ Meeting 2001 International Congress on FEM Technology
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Fig. 16 Mesh smoothing dialog The smoothed finite element mesh is shown in Fig. 17. The Von Mises stress distribution in the obtained structure is shown in Fig. 18. As is to be seen, undesirable stress peaks occur. These shall be eliminated in the following step via biological growth.
Fig. 17: Smoothed structure.
Fig. 18: Von Mises stress distribution in the resulting structure.
Fig. 19 shows the necessary user interaction, choosing the number of iterations and a scaling constant.
Fig. 19 Biological growth using pseudo-thermal strains
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In Fig. 20, the resulting Von Mises stress distribution after 3 steps of biological growth is shown in the same scale as in Fig. 18. The last step to perform is the export of the faceted finite element surfaces in any CAD software using STL. Fig. 21 shows the surface model of the optimized solid component.
Fig. 20: Resulting structure with Von Mises stress Fig. 21: STL based CAD export of the faceted after biological growth. surface of the solid component.
Conclusions The presented software program TopoSLang is a very efficient tool for the topology and shape optimization of solid components like massive tools. The software development focused to combine sophisticated mathematical algorithms with a user-friendly GUI to support a continuous workflow. At the moment the alpha development phase is completed. Within a short time a first commercial version will be available via CAD-FEM. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Bayer V., Bucher C. et al.: “SLang - the Structural Language Version 4.2”, Institute of Structural Mechanics - Bauhaus-University Weimar August 2, 2001. Bendsoe M., Kikuchi N.: “Generating optimal topologies in structural design using a homogenization method”, Comp. Meth. Appl. Mech. Engng., 71, pp. 197-224. Bucher C.G., H.J. Pradlwarter, G.I. Schuëller.: „COSSAN - Ein Beitrag zur SoftwareEntwicklung für die Zuverlässigkeitsbewertung von Strukturen“, VDI-Bericht Nr. 771, Düsseldorf, Germany, 1989, pp 271 - 281, 1989. Bucher C.G., G.I. Schuëller.: „Software for Reliability Based Analysis”, Structural Safety 16, pp 13 - 22, 1994. Bucher C., Y. Schorling, W. Wall.: “SLang - the Structural Language, a tool for computational stochastic structural analysis”, In S. Sture, Editor, Proc. 10th ASCE Eng. Mech. Conf., Boulder, CO, May 21-24, 1995, pages 1123 -1126. ASCE, New York, 1995. Kohn R. V., Strang G.: “Optimal design and relaxation of variational problems i-iii”, Comm. Pure Appl. Math., 39, pp. 113-137, 139-182, 353-377, 1986. Mattheck C.: “Engineering components grow like trees”, Mat.-wiss. U. Werk- sto tech., (21):143168, 1990. Maute K., S. Schwarz, E. Ramm: „Structural optimization - the interaction between form and mechanics”, ZAMM - Zeitschrift fý ur Angewandte Mathematik und Mechanik, 79(10):651 - 673, 1999. Mlejnek H. P.: “An engineering approach to optimal material distribution and shape finding”, Comp. Meth. Appl. Mech. Engng., 106:1-26, 1993. Vogel F.: “Topology optimization of linear-elastic structures with Ansys 5.4”, In Proc. NAFEMS Conference on Topology Optimization, 23 Sept. 1997. Vogel F.: “Topology Optimization API - a User's Manual”, inuTech GmbH, 2000. Zillober C.: “Software manual for SCPIP 2.2”, Technical Report TR01-2, Informatik, Universität Bayreuth. January 2001. Bucher C., Will J., Riedel J.: Multidisciplinary non-linear optimization with Optimizing Structural Language OptiSLang, 19.CAD-FEM User’s Meeting, Potsdam, 2001.
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