Topology optimization with adaptive mesh refinement
Eric de Sturler
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Abstract
We outline a robust method for topology optimization with adaptive mesh refinement and derefinement (AMR). Since the total volume fraction in topology optimization is usually modest, after a few initial iterations the domain of computation is largely void. It is inefficient to have many small elements in such regions, as these contribute significantly to the overall computational cost but little to the accuracy of computation and design. At the same time, we want high spatial resolution for accurate three-dimensional designs to avoid significant postprocessing or interpretation. AMR offers the possibility to balance these two requirements, but it has received little attention in the context of topology optimization. We will discus approaches by Costas and Alves [2] and Stainko . Unfortunately, both approaches may lead to suboptimal designs that are mesh dependent. We extend these approaches to obtain a method that yields optimal designs, and we show experimentally that our improvements lead to designs that are equivalent to designs computed on uniform meshes at the finest level of refinement. Furthermore, we demonstrate significant reductions of run time by using AMR and efficient methods for the solution of the resulting large, linear systems, following Wang et al. [4].
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arXiv (Cornell University), 2010
We present an improved method for topology optimization with both adaptive mesh refinement and derefinement. Since the total volume fraction in topology optimization is usually modest, after a few initial iterations the domain of computation is largely void. Hence, it is inefficient to have many small elements, in such regions, that contribute significantly to the overall computational cost but contribute little to the accuracy of computation and design. At the same time, we want high spatial resolution for accurate three-dimensional designs to avoid postprocessing or interpretation as much as possible. Dynamic adaptive mesh refinement (AMR) offers the possibility to balance these two requirements. We discuss requirements on AMR for topology optimization and the algorithmic features to implement them. The numerical design problems demonstrate (1) that our AMR strategy for topology optimization leads to designs that are equivalent to optimal designs on uniform meshes, (2) how AMR strategies that do not satisfy the postulated requirements may lead to suboptimal designs, and (3) that our AMR strategy significantly reduces the time to compute optimal designs.
A simple adaptive mesh refinement scheme for topology optimization using polygonal meshes
Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2018
The objective of topology optimization is to find the most efficient distribution of material (optimal topology) in a given domain, subjected to design constraints defined by the user. The quality of the new boundary representation depends on the level of mesh refinement: The greater the number of elements in the mesh, the better is the representation of the optimized structure. However, the use of refined meshes implies a high computational cost, particularly with regard to the numerical solution of the linear systems of equations that arise from the finite element method. This paper proposes a new computational strategy for adaptive local mesh refinement using polygonal finite elements in arbitrary two-dimensional domains. The idea is to perform a mesh refinement in regions of material concentration, mostly in inner and outer boundaries, and a mesh derefinement in regions of low material concentration, such as in internal holes. Thus, it is possible to obtain high-resolution optimal topologies with a relatively low computational cost. Representative examples are presented to demonstrate the robustness and efficiency of the proposed methodology by comparing the results obtained here with results from the literature where refined meshes are kept constant throughout the topology optimization process.
New Advances in Topology Optimization
2008
We outline a robust method for topology optimization with adaptive mesh refinement and derefinement (AMR). Since the total volume fraction in topology optimization is usually modest, after a few initial iterations the domain of computation is largely void. It is inefficient to have many small elements in such regions, as these contribute significantly to the overall computational cost but little to the accuracy of computation and design. At the same time, we want high spatial resolution for accurate three-dimensional designs to avoid significant postprocessing or interpretation. AMR offers the possibility to balance these two requirements, but it has received little attention in the context of topology optimization. We will discus approaches by Costas and Alves [2] and Stainko [3]. Unfortunately, both approaches may lead to suboptimal designs that are mesh dependent. We extend these approaches to obtain a method that yields optimal designs, and we show experimentally that our improvements lead to designs that are equivalent to designs computed on uniform meshes at the finest level of refinement. Furthermore, we demonstrate significant reductions of run time by using AMR and efficient methods for the solution of the resulting large, linear systems, following Wang et al. [4]. 6th International Conference on Computation of Shell and Spatial Structures IASS-IACM 2008, Ithaca not satisfy these properties. We propose simple but essential changes to these methodologies that lead to AMR based designs that are equivalent (up to some small tolerance) to designs on uniform fine meshes. In topology optimization we solve for the material distribution in a given design domain . Here, we minimize the compliance of a structure under given loads as a function of the material distribution. To solve this problem numerically, we discretize the computational domain using finite elements, where we use a lower order interpolation for the density field (material distribution) than for the displacement field. We take the most common approach using trilinear interpolation for the displacement field and constant density in each element. The compliance minimization problem after finite element discretization is defined as Ω [ ] ( ) 0 ,1
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International Journal for Numerical Methods in Engineering, 2009
In topology optimization literature, the parameterization of design is commonly carried out on uniform grids consisting of Lagrangian-type finite elements (e.g. linear quads). These formulations, however, suffer from numerical anomalies such as checkerboard patterns and one-node connections, which has prompted extensive research on these topics. A problem less often noted is that the constrained geometry of these discretizations can cause bias in the orientation of members, leading to mesh-dependent sub-optimal designs. Thus, to address the geometric features of the spatial discretization, we examine the use of unstructured meshes in reducing the influence of mesh geometry on topology optimization solutions. More specifically, we consider polygonal meshes constructed from Voronoi tessellations, which in addition to possessing higher degree of geometric isotropy, allow for greater flexibility in discretizing complex domains without suffering from numerical instabilities. Copyright © 2009 John Wiley & Sons, Ltd.
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Computers & Structures, 2019
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Towards adaptive topology optimization
Advances in Engineering Software, 2016
This paper presents a new fully-automated adaptation strategy for structural topology optimization (TO) methods. In this work, TO is based on the SIMP method on unstructured tetrahedral meshes. The SIMP density gradient is used to locate solid-void interface and hadaptation is applied for a better definition of this interface and, at the same time, de-refinement is performed to coarsen the mesh in fully solid and void regions. Since the mesh is no longer uniform after such an adaptation, classical filtering techniques have to be revisited to ensure mesh-independency and checkerboard-free designs. Using this adaptive scheme improves the objective function minimization and leads to a higher resolution in the description of the optimal shape boundary (solid-void interface) at a lower computational cost. This paper combines a 3D implementation of the SIMP method for unstructured tetrahedral meshes with an original mesh adaptation strategy. The approach is validated on several examples to illustrate its effectiveness.
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Structural and Multidisciplinary Optimization, 2010
This paper presents a multiresolution topology optimization (MTOP) scheme to obtain high resolution designs with relatively low computational cost. We employ three distinct discretization levels for the topology optimization procedure: the displacement mesh (or finite element mesh) to perform the analysis, the design variable mesh to perform the optimization, and the density mesh (or density element mesh) to represent material distribution and compute the stiffness matrices. We employ a coarser discretization for finite elements and finer discretization for both density elements and design variables. A projection scheme is employed to compute the element densities from design variables and control the length scale of the material density. We demonstrate via various two- and three-dimensional numerical examples that the resolution of the design can be significantly improved without refining the finite element mesh.
High-order Finite Elements for Topology Optimization
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ArXiv, 2020
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International Journal for Numerical Methods in Engineering, 2012
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Le Centre pour la Communication Scientifique Directe - HAL - memSIC, 2021
Topology optimization is devoted to the optimal design of structures: It aims at finding the best material distribution inside a working domain while fulfilling mechanical, geometrical and manufacturing specifications. Conceptually different from parametric or size optimization, topology optimization relies on a freeform approach enabling to search for the optimal design in a larger space of configurations and promoting disruptive design. The need for lighter and efficient structural solutions has made topology optimization a vigorous research field in both academic and industrial structural engineering communities. This contribution presents a Research and Development software platform for shape and topology optimization where the computational process is carried out in a level set framework combined with a body-fitted approach.
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Structural and Multidisciplinary Optimization, 2017
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