Filter techniques in shape optimization with CAD-free parametrization

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1 6 th World Congresses of Structural and Multidisciplinary Optimization Rio de Janeiro, 30 May - 03 June 2005, Brazil Filter techniques in shape optimization with CAD-free parametrization F. Daoud 1, M. Firl 2, K.-U. Bletzinger 3 daoud@bv.tum.de 1, firl@bv.tum.de 2, kub@bv.tum.de 3, Lehrstuhl für Statik, TU München, Arcisstr. 21, D München, Germany 1,2,3 1. Abstract In this contribution an innovative method for shape optimization with parametrization directly on FE mesh is proposed. The major shortcomings of CAD-free parametrization are discussed and global filter techniques are presented in this context. A global filter for shape optimization with FE-parametrization is developed and applied to numerical examples of basic shapes. The influence of locking phenomena in the context of shape optimization is discussed. Finally a local filter for bead optimization is proposed. 2. Keywords: shape optimization, CAD-free parametrization, FE-parametrization, filter techniques, bead optimization, locking. 3. Introduction Parametrization of the design for shape optimization can be based on CAD models or directly on the FE mesh (CAD-free parametrization). In CAD based design optimization, design variables are chosen among the parameters related to the underlying CAD model. During the optimization process, the update of the finite element mesh is due to changes of the CAD model. In FE based design optimization, nodal coordinates are taken as design variables. This parametrization gives more freedom to the optimization process, since the result of the optimization problem is not restricted to the preselected CAD design space. However, there are many technical facts which in the past made this natural choice of optimization model to appear as prohibitive, as there are: the large number of design variables, the wiggly shape and the inherent problems of FE discretization (e.g. locking) which may lead to mechanically wrong answers. This contribution presents an innovative approach to overcome these shortcomings by a combination of several techniques which results in a powerful and flexible tool for the preliminary design of free form shells. The problem of the large number of variables is overcome by the use of adjoint sensitivity analysis. From the mechanical point of view locking-free elements (shell elements) are indispensable. Not only the correct system response depends on such element formulation, but also the feasible domain of optimization problem is severely affected by locking phenomena, c.f. Camprubi et al. [7], Bischoff et al. [5]. Moreover, it can be shown, that locking phenomena are responsible for shape oscillations on element level, which is the smallest scale that can be represented by a chosen discretization. The main problem of the FE-based design is that wiggly shapes may be obtained as optimal designs. These wiggles are mesh dependent and obviously it is not desirable that the optimum design depends on the coarseness of the mesh used. Therefore, the aim of this research is to provide a tool to control the smoothness of the designs obtained as result of shape optimization and to avoid mesh dependency. Based on the filter proposed by Sigmund in [12] to overcome checkerboard and mesh dependency problems in topology optimization, a filter for shape optimization is proposed. Through this sensitivity filter, not only high frequency waves are smoothed, but also the global wave length of the final design can be controlled by adjusting the filter radius. 4. Filter techniques In general, a shape can be understood as the superposition of several principal shape functions each characterized by a certain wave length. Results of shape optimization in FE based design often of local nature are generated by arbitrary combinations of principal shape functions. As a consequence, successful optimization methods must be able to keep control of the desired mesh independent wave lengths of the resulting shapes by means of filter techniques. Nowadays filter techniques are well established procedures in topology optimization and in many other disciplines, c.f. Bendsoe and Sigmund [3], Bängtsson et al. [2]. These methods are by far more than just pragmatical tools, which do a good job in practice. The research in the last two decades especially in mathematics gives us many technically 1

2 mature methods (for more details see Tikhonov et al. [13]). The above mentioned problem of high-frequency waves in the optimization results arises from the many possible scales of the analyzed design. On the one hand many local minima exist, on the other hand during the optimization process long waves of possible solutions compete with divers short waves or wiggles (of other scales). Exactly at this point the filter techniques are engaged to selectively affect the desired wave length and to filter it out. The aim is to develop a procedure, by which smoothness and curvature of the optimized structures are controlled, involving some user-defined geometrical measurement of the modeled structure. The influence of the high-frequency waves, also called disturbance or noise, can be demonstrated on the basis of the one-dimensional case. A cut-out (Figure 1) shows a long wave as well as other superposed short ones. Figure 1: Influence of high-frequency waves on gradients If the oscillating part of the solution is undesirable, then it is obvious that the wiggles (the superposed short waves) influence the sensitivity severely, e.g. at Point P in Figure 1 we notice even a change of sign. It is obvious that the sensitivity of the undisturbed curve deviates from the value of the disturbed one, which definitely affects the whole optimization procedure, exceedingly in gradient based algorithms. The task now is to get rid of the noisy wiggles systematically. An idea how to control waves in various scales in a noisy system response can be borrowed from signal technologies, where the waves, before transmission, are transformed by means of Fourier transformation or wavelet transformation. It is common technique to approximate a function f by superposing periodical shape functions N i (x) and the corresponding amplitudes or function values f i, respectively f(x) f(x) = i N i (x) f i (1) Beside the shape functions, there are also the dual functions D i (x), which do the opposite job, namely filtering the values f i out. The dual function is defined as follows 1 N i (ξ) D j (ξ) dω = δ ij (2) Ω Ω(x) The integration domain Ω depends on the used shape and dual functions. It usually represents the smallest wave length described in the approximation. If we apply the dual function on the approximation f(x), the evaluations of the original function at the sampling points f i are exactly filtered out. On the other hand, if the dual function is applied on the original (noisy) function f(x), the oscillations, which cannot be represented by the shape functions, will be eliminated, as their wave length is smaller than the radius of Ω. A smoothed function evaluation f m is produced. 1 f(ξ) D(ξ) dω = f m (x) (3) Ω Ω(x) 2

3 The integration, Eq (3), over the domain Ω can be interpreted as a way to determine the weighted mean value at each sampling point, which is usually located in the center of the integration domain. This issue motivates a smoothing method for shape optimization. The choice of a suitable dual function and the discretization of Eq (3), where the sensitivities are filtered instead of the function itself, yields f m x k e = n i=1 D i f x k i n (4) D i n number of nodal points inside the integration domain Ω (including the center e) D i = D(R R ei ) weighting function concerning spacial and topological distance R smoothing radius representing Ω R ei Distance between sampling point e (the center) and node i. k iterator for variable coordinates at the node (e.g. X, Y and Z). i=1 This approach is applied in a similar manner to topology optimization, c.f. Bendsoe and Sigmund [3]. Figure 2: spacial (geometrical) and topological distance. Figure 3: element and node levels arround the center. The radius of Ω controls the smallest desired wave length (size of wiggles) in the optimized structure. An additional difficulty in shape optimization concerns the evaluation of distances on the curved surfaces, as the spacial distance only is not enough to determine the correct weighting factors in the sphere of influence of the filter radius. This fact can be demonstrated by means of an intersection curve on a curved surface (Figure 2). Taking point A as a sampling point (point at which the sensitivity is filtered), we notice that node 1, from the topological point of view, is closer to node A than node 2, and accordingly node 1 should possess a greater weighting factor. However, if only the direct distance is considered, then node 2 will get the bigger weight. One possibility is to measure the distances on the curved surface, which is difficult to perform, particularly for discretized structures. To avoid measuring on the curved surface, we introduce a hierarchical weighting system. The FE nodes around the sampling point are classified in levels. Each level gets a special topological weighting factor, as depicted in Figure 3. The final weighting factor is a combination including both information the topological distance and the spacial one. In this way we keep the algorithmic effort low, by scanning the a priori created node map (in the input phase) within the filter radius. As a result of the filtering process the smallest (shortest) waves in the optimized structures are not controlled by discretization parameters (the size of the finite elements), but rather by user defined measurements like the filter radius. 5. Numerical examples The effects of the presented filter will be demonstrated on several exemplary but meaningful benchmarks of minimizing strain energy (compliance). The examples deal with some basic shapes to show the shortcomings of CAD-free shape optimization without filter techniques. The comparison with the results after applying the proposed filter shows the benefits of the method. The following examples are restricted to the static and linear case. Furthermore a linear elastic material law is applied. 3

4 (a) Example 1 (b) Example 2 (c) Example 3 Figure 4: Start geometry (a) Example 1 (b) Example 2 (c) Example 3 Figure 5: Optimization results without filter (a) Example 1 (b) Example 2 (c) Example 3 Figure 6: Optimization results with filter radius equals half the span length In the first example a Navier-supported circular plate under self-weight is optimized. Since the exact plane (horizontal) position as a start geometry results in a maximum of strain energy, as the whole energy is produced in terms of bending, some disturbance in form of start deflection is necessary. The problems involves 294 design variables which are the Z-coordinates (vertical movement) of all nodes except for the supported ones. As the state of minimal strain energy is reached when the load is carried by membrane forces, we expect a shape which has a tendency towards a minimal surface. Locking-free DSG linear shell elements are used, c.f. Bletzinger et al. [6]. The sensitivity analysis is performed by the adjoint method which reduces the calculation cost severely [10]. In the second example a Navier-supported square plate under point loading in the center is optimized. The same remark about start geometry applies also in this case. For parametrization of the geometry 436 design variables are used. In the third example the start geometry is a rectangular container under internal pressure, where we expect a cylindrical shape as a solution of optimization. In this example 438 design variables are used. The optimization results in Figure 6 show the performance of the proposed method. The wiggles of wave lengths smaller than the filter ra4

5 dius (half the span length) are eliminated, as they cannot be represented by the chosen shape functions. The smoothness of the resulting structures is controlled by means of a user-defined parameter derived from the optimized model (filter radius) and is mesh independent which is the main aim of this approach. 6. Locking in the context of shape optimization Locking phenomena are practically as old as the finite element method itself. Already in the early days of FEM the existance of locking and its influence on the results of structural analysis was noticed and investigated [1, 4]. However, the effect of the various locking phenomena on the optimization process is subject of actual research. feasible domain (Locking) feasible domain (Locking-free) VF Optimum (Locking) Optimum (Locking-free) Figure 7: Influence of locking on the feasible domain. The problems of displacement elements, or any other finite element formulation suffering from locking are presented in the following. In [5] the influence of locking on the formulation of the optimization problem is studied, with the focus on the consequences for the feasible domain. It can be observed, that every function and functional depending on the strain energy and/or the vector of nodal displacements, contain a defect due to locking. As a consequence, any approach based on such an element formulation will include the same phenomena. It can be shown, that locking modifies the functions involved in optimization process, especially stress constraints, which restrict the search space, so that the feasible domain is severely affected [8]. Changes in the feasible domain can cause infeasible optimization results as can be seen exemplarily in Figure 7. The irregular influence of locking on the structural response, particularly on the sensitivities, is responsible for the highfrequency oscillations the wiggles of smallest scale on element level in the results of shape optimization of freeform shells. Due to locking the nodal displacements (as state variables of structural analysis) are always underestimated, however the derived fields as strains and stresses show the typical oscillations on element level (see Figure 8). These oscillations are reproduced in the sensitivity analysis, that provides the gradients for gradient-based optimization algorithms. This inherent waviness enhances the wiggly shapes during optimization. The result of shape optimization of a Naviersupported square plate under uniform surface load is depicted in Figure 9. The result of optimization with displacement elements (Figure 9(a)) manifests the fact, that the oscillating structural response on element level due to locking leads to wiggles in the same scale. The usage of locking-free DSG shell elements however, cancels the wiggles on element level (Figure 9(b)). The wave control on system level The main locking phenomenon investigated in this work is the transverse shear locking. Objective function: minimization of strain energy. Discrete Strain Gap [6, 11]. 5

6 must be performed additionally e.g. by means of the presented filter techniques. (a) Distribution of shear forces for displacement elements. (b) Distribution of shear forces for DSG elements. Figure 8: Influence of locking on shape optimization. (a) Optimization with displacement elements (without filter techniques). (b) Optimization with DSG elements (without filter techniques). Figure 9: Influence of locking on waviness of the solution. 7. Bead optimization The proposed filter technique represents a global filter, which operates structure-wide to homogenize the structural response. It also defines an interface between the designer and the optimized structure, where the scales of waves, which should remain in the optimization process, can be determined independent of mesh coarseness. The basic idea of the bead optimization algorithm is to localize the influence of filter to special regions, the so-called hot spots, where local shape modifications improve the mechanical properties of the structure significantly. These special regions form the bead bodies during optimization. To determine the position and orientation of the hot spots a suitable bead criterion is needed. In [9] Emmrich suggests to place the beads along the trajectories of principle moments. In this work a generic approach basing on an appropriate Sensitivity analysis is applied to achieve the desired improvement by means of local shape alterations. In the case of the minimization of strain energy as objective function the projection of the their sensitivities on the director vector at each FE node is taken as a mechanichal quantity to determine, if 6

7 a FE node belongs to the bead body or not. After sensitivity analysis the gradients are denoised with a global filter and sorted in descending order to choose the candidates for building the bead centers. At each candidate for a bead center the magnitude of the projected sensitivity at the node must exceed a specified limit. By experience the follwoing value s f lim provides a good reference: f s f lim s i s f lim = s f mean + 1 n s f mean = 1 n n i=1 n i=1 ( f s i s f mean ) 2 f s i (5) (a) 1. Iteration. (b) 2. Iteration. (c) After bead fusion. Figure 10: Bead optimization of a cantilever plate (clamped on the right hand side). From the sorted list of candidates the node on top (the highest sensitivity) is marked as the first bead center. The remaining candidates must additionally satisfy the following condition, bevor they can declared as addtional centers: Dist(Cen j, Cand i ) 2αR b (6) with Cenj FE nodes, which are already marked as bead centers Cand i analysed candidate Dist(Cen j, Cand i ) distance between the node Cand i and all known centers Cen j α interaction factor R b influence radius of the bead. The bead optimization of a cantilever plate (Figure 10) shows the functionality of the proposed local filter. The convergence in this case is achieved, when the bead pattern converges, i.e. a stable pattern with no more bead centers is maintained. In the example of the clamped plate convergence is gotton after two iterations. After convergence the structure consists of many separate contiguous beads (Figure 10(b)). The question arises, which of the fugues between the beads should better removed and which should stay. In all presented numerical results the obective is the minimization of strain energy. 7

8 (a) Start Geometry. (b) Stresses in the Start Geometry. (c) Stresses in bead structure. Figure 11: Bead optimization of an uniaxial plate. The answer is attained by an additional sensitivity analysis for the FE nodes lying on the boundary between to beads. The projection of the sensitivity information (of fugue nodes) on the director vector and the comparison with the sensitivities at the involved bead centers provides a mechanical quatity to decide, if the boundary is to be removed or not, as can be seen in Figure 10(c). In the Figures 11 and 12 some numerical examples show the application and effect of the proposed bead optimization algorithm based on filter techniques and model paramertization directly on the FE mesh. (a) Sideview. (b) Topview. (c) Perspective view. 8. References Figure 12: Bead optimization of a container subjected to internal pressure. [1] I. Babuška and M. Suri. On locking and robustness in the finite element method. Technical Report BN-1112, Inst. for Physical Sci and Tech., University of Maryland, College Park Campus, May [2] Erik Bängtsson, Daniel Noreland, and Martin Berggren. Shape optimization of an acoustic horn. Computer methods in applied mechanics and engineering, (192): , [3] M. P. Bendsøe and O. Sigmund. Topology optimization. Theory, methods and applications. Springer Verlag Berlin, Heidelberg, New York,

9 [4] M. Bischoff. Theorie und Numerik einer dreidimensionalen Schalenformulierung. PhD thesis, Institut für Baustatik, Universität Stuttgart, [5] Manfred Bischoff, Natalia Camprubi, and Kai-Uwe Bletzinger. Shape optimization of shells and locking. Computers & Structures, (82): , [6] K.-U. Bletzinger, B. Bischoff, and E. Ramm. A unified approach for shear-locking free triangular and rectangular shell finite elements. Computers & Structures, (75): , [7] N. Camprubi, M. Bischoff, and F. Koschnick. High quality structural response and sensitivity analysis in shape optimal design of shells. In WCSMO 5th World Congress of Structural and Multidisciplinary Optimization, Lidodi Vesolo - Venice, 19 May [8] Natalia Camprubi. Design and Analysis in Shape Optimization of Shells. PhD thesis, Lehrstuhl für Statik, Fakultät für Bauingenieur- und Vermessungswesen, Technische Universität München, [9] Dieter Emmrich. Methodische Sickenentwicklung. PhD thesis, Institut für Fahrzeugtechnik,Technische Hochschule Karlsruhe, [10] S. Kimmich. Strukturoptimierung und Sensibilitätsanalyse mit finiten Elementen. PhD thesis, Institut für Baustatik der Universität Stuttgart, [11] Frank Koschnick. Geometrische Locking-Effekte bei Finiten Elementen und ein allgemeines Konzept zu ihrer Vermeidung. PhD thesis, Lehrstuhl für Statik, Fakultät für Bau- und Vermessungswesen der Technischen Universität München, [12] O. Sigmund and P. Peterson. Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Structural Optimization, (16):68 75, [13] A. N. Tikhonov, A. V. Stepanov, and A. G. Yagola. Numerical methods for the solution of ill-posed problems. Kluwer Academic Publishers,