Density-based Topology Optimization Methods Research Articles

Coupling of bond-based peridynamics and continuous density-based topology optimization methods for effective design of three-dimensional structures with discontinuities

This study proposes continuous density-based three-dimensional topology optimization (TO) approaches developed by coupling the peridynamic theory (PD) with optimality criteria (OC) and proportional approach (PROP). These frameworks, abbreviated as PD-OC-TO and PD-PROP-TO, can be practically utilized to enhance the fracture toughness of the structures during the optimization process by taking critical regions into account as pre-defined cracks. Breaking the non-local interactions (bonds) between relevant PD particles enables us to readily model cracks. Utilizing this advantage, we solve several benchmark optimization problems including different numbers, positions, and alignments of the cracks. The major differences between the proposed methods are examined by comparing optimum topologies for various cracked scenarios. Moreover, the mechanical behaviour of the optimized structures is investigated under dynamic loads to prove the significant improvements achieved by the present approach in the final designs. The results of dynamic analyses reveal the viability of both PD-TO methods for increasing the fracture toughness of the structure in the optimization stage. Overall, the proposed approach is confirmed as a superior design and optimization tool for future engineering structures.Graphical abstract

Topology optimization of thermal cloaks in euclidean spaces and manifolds using an extended level set method

Thermal cloaks are devices designed to shield an object against thermal detection, which have attracted growing interest in research. This paper proposes to design thermal cloaks using the level-set-based shape and topology optimization in the context of pure heat conduction. The cloaking effect is achieved by optimizing the distribution of two bulk heat conductive materials to eliminate the temperature disturbance induced by the introduction of the insulator (cloaking region) into a homogeneous thermal conduction medium. The optimized thermal cloaks are free of high anisotropy and nonhomogeneity commonly seen in the popular transformation thermotics or scattering cancellation methods. Due to the clear boundary characteristic of the level set representation, no sophisticated filtering techniques are required to suppress the appearance of ”gray regions” as opposed to the density-based topology optimization methods. Considering the fact that the device components that need to be thermally cloaked, e.g., sensors, can take an arbitrary free-form shape, a conformal thermal cloak on the manifold is also topologically optimized using the extended level set method (X-LSM), which has not been reported in the literature. The structural boundary is evolved by solving the (modified) Hamilton-Jacobi equation. The feasibility and robustness of the proposed method to design thermal meta-devices with cloaking functionality are demonstrated through a number of 2D and 3D (solid and shell) numerical examples with different cloaking regions (circular, human-shaped, spherical, and curved circular). This work may shed light on further exploration of the thermal meta-devices in the heat flux manipulation regime.

Improvement of a topological level-set approach to find optimal topology by considering body forces

PurposeBody forces are always applied to structures in the form of the weight of materials. In some cases, they can be neglected in comparison with other applied forces. Nevertheless, there is a wide range of structures in civil and mechanical engineering in which weight or other types of body forces are the main portions of the applied loads. The optimal topology of these structures is investigated in this study.Design/methodology/approachTopology optimization plays an increasingly important role in structural design. In this study, the topological derivative under body forces is used in a level-set-based topology optimization method. Instability during the optimization process is addressed, and a heuristic solution is proposed to overcome the challenge. Moreover, body forces in combination with thermal loading are investigated in this study.FindingsBody forces are design-dependent loads that usually add complexity to the optimization process. Some problems have already been addressed in density-based topology optimization methods. In the present study, the body forces in a topological level-set approach are investigated. This paper finds that the used topological derivative is a flat field that causes some instabilities in the optimization process. The main novelty of this study is a technique used to overcome this challenge by using a weighted combination.Originality/valueThere is a lack of studies on level-set approaches that account for design-dependent body forces and the proposed method helps to understand the challenges posed in such methods. A powerful level-set-based approach is used for this purpose. Several examples are provided to illustrate the efficiency of this method. Moreover, the results show the effect of body forces and thermal loading on the optimal layout of the structures.

Simultaneous material, shape and topology optimization

Using three design fields we develop an optimization environment that can simultaneously optimize material, shape and topology. We use the implicit representation of the boundaries with level-set functions that define the shape and topology. Differentiable R-functions allow us to combine these shapes and topology descriptions with Boolean operations. Additionally, we incorporate design dependent-stiffness materials with another design field. Notably, this framework accommodates design dependent loads, has the ability to introduce holes, and ensures the satisfaction of optimality criteria. It builds upon the fictitious domain, ersatz material, material interpolation and level-set methods. It also borrows from parameterized density-based topology optimization methods. Since analytical sensitivities can be computed, we use efficient nonlinear programming algorithms to update the design instead of the Hamilton–Jacobi’s scheme of level-set methods. We illustrate the features of our framework by designing a cantilever beam with octet truss microlattice, a dam with design-dependent loads, and a composite clevis plate.

Continuous density-based topology optimization of cracked structures using peridynamics

Peridynamics (PD) is a meshless approach that addresses some of the difficulties and limitations associated with mesh-based topology optimization (TO) methods. This study investigates topology optimization of structures with and without embedded cracks using peridynamics (PD-TO). To this end, PD is coupled with two different continuous density-based topology optimization methods, namely the optimality criteria and the proportional optimization. The optimization results are compared for a continuous definition of the design variables, which are the relative densities defined for PD particles. The checkerboard issue has been removed using filtering schemes. The accuracy of the proposed PD-TO approach is validated by solving benchmark problems and comparing the optimal topologies with those obtained using a FEM-based topology optimization. Various problems are solved with and without defects (cracks) under different loading and constraint boundary conditions. Topology optimization for an unstructured discretization problem has also been investigated applied to a complex geometry. The optimal topology of a cracked structure may change for different optimization methods. The numerical results demonstrate the accuracy, high efficiency, and robustness of the PD-TO approach.

De-homogenization of optimal multi-scale 3D topologies

This paper presents a highly efficient method to obtain high-resolution, near-optimal 3D topologies optimized for minimum compliance on a standard PC. Using an implicit geometry description we derive a single-scale interpretation of optimal multi-scale designs on a very fine mesh (de-homogenization). By performing homogenization-based topology optimization, optimal multi-scale designs are obtained on a relatively coarse mesh resulting in a low computational cost. As microstructure parameterization we use orthogonal rank-3 microstructures, which are known to be optimal for a single loading case. Furthermore, a method to get explicit control of the minimum feature size and complexity of the final shapes will be discussed. Numerical examples show excellent performance of these fine-scale designs resulting in objective values similar to the homogenization-based designs. Comparisons with well-established density-based topology optimization methods show a reduction in computational cost of 3 orders of magnitude, paving the way for giga-scale designs on a standard PC.

A material-field series-expansion method for topology optimization of continuum structures

This paper proposes a new topology optimization method based on a material-field topology description and a reduced series expansion. Not only does the proposed method greatly reduce the number of design variables, it also inherently avoids checkerboard patterns and the mesh dependency of conventional density-based topology optimization methods. In the present method, the structural topology is represented by a bounded material field with spatial dependency. The correlation length is introduced here to control the length-scale of the topology distribution. After approximating the bounded material field as a linear function of a reduced set of undetermined coefficients by using a series expansion, the topology optimization problem is constructed in the form of finding the optimal coefficients of the material field with the minimum structural compliance. A standard, gradient-based algorithm incorporating the design sensitivity information is then used to solve the optimization problem effectively. Several examples are given to demonstrate the effectiveness and applicability of the proposed topology optimization method.

Topology optimization of piezo modal transducers considering electrode connectivity constraints

Piezo modal sensors/actuators are devices that utilize the piezoelectric effect to measure/excite a particular mode while remaining insensitive to the non-targeted modes. The design of such transducers is typically confined to determining the location of tailor-made electrodes of opposite polarity, meaning one seeks to determine which electrode areas have positive polarity (+1) and which areas are negative (-1). Surprisingly, these electrode profiles can be obtained as the solutions of a linear programming problem that elegantly models the design problem. Designs resulting from this process, however, typically exhibit isolated features of like-polarity, and this characteristic typically becomes more pronounced as additional modes are considered. These isolated features pose significant challenges from a manufacturing and operation point of view, as each feature must be connected to a current source, thereby resulting in complex or cumbersome wiring schemes. This work introduces a connectivity constraint on the electrode profiles to dramatically simplify electrode phase connectivity and therefore the required wiring schemes. The resulting design problem is solved using density-based topology optimization methods and numerical examples confirm that piezo modal transducers can be designed with simply-connected electrode phases, thereby providing efficient performance together with improved manufacturability.

Evolutionary topology optimization for structural compliance minimization considering design-dependent FSI loads

This work presents an extended bi-directional evolutionary structural optimization (BESO) method applied to static structural design problems considering the interaction between viscous fluid flows and linearly elastic structures. The fluid flow is governed by incompressible and steady-state Navier-Stokes equations. Both domains are solved with the finite element method and simplifying conditions are assumed for the fluid-structure coupling, such as small structural displacements and deformations in a staggered method. The presented BESO method aims to minimize structural compliance in a so called “wet” optimization problem, in which the fluid loads location, direction and magnitude depend on the structural layout. In this type of design-dependent loading problem, density-based topology optimization methods require extra numerical techniques (usually mixed models with overlapping domains) in order to model the interaction of different governing equations during the optimization procedures. In this work, the discrete nature of the evolutionary topology optimization approach allows the fluid-structure boundaries to be modelled and modified straightforwardly by switching the discrete design variables between fluid and structural finite elements. Therefore, separate domains are used in this approach. Numerical results show that the BESO-based methods can be applied to this kind of multiphysics problem effectively and efficiently.

Computational multiobjective topology optimization of silicon anode structures for lithium-ion batteries

This study utilizes computational topology optimization methods for the systematic design of optimal multifunctional silicon anode structures for lithium-ion batteries. In order to develop next generation high performance lithium-ion batteries, key design challenges relating to the silicon anode structure must be addressed, namely the lithiation-induced mechanical degradation and the low intrinsic electrical conductivity of silicon. As such this work considers two design objectives, the first being minimum compliance under design dependent volume expansion, and the second maximum electrical conduction through the structure, both of which are subject to a constraint on material volume. Density-based topology optimization methods are employed in conjunction with regularization techniques, a continuation scheme, and mathematical programming methods. The objectives are first considered individually, during which the influence of the minimum structural feature size and prescribed volume fraction are investigated. The methodology is subsequently extended to a bi-objective formulation to simultaneously address both the structural and conduction design criteria. The weighted sum method is used to derive the Pareto fronts, which demonstrate a clear trade-off between the competing design objectives. A rigid frame structure was found to be an excellent compromise between the structural and conduction design criteria, providing both the required structural rigidity and direct conduction pathways. The developments and results presented in this work provide a foundation for the informed design and development of silicon anode structures for high performance lithium-ion batteries.

A geometry projection method for continuum-based topology optimization with discrete elements

Abstract This article describes a method for the continuum-based topology optimization of structures made of discrete elements. In particular, we examine the optimization of linearly elastic planar structures made of bars of fixed width and semicircular ends. The design space for the optimization consists of the endpoint locations of the bar’s medial axes and their out-of-plane thicknesses. To circumvent re-meshing upon design changes, we project the design onto a fixed analysis grid using a differentiable geometry projection that results in a density field indicating the fraction of solid material anywhere in the design space, as in density-based topology optimization methods. The out-of-plane thickness is penalized so that the optimizer is capable of removing bars during the optimization. The differentiability of the projection allows for the computation via the chain rule of design sensitivities of responses of interest, and therefore it allows for the use of robust and efficient gradient-based optimization methods. Notably, this approach makes it easier to fabricate optimal designs by using off-the-shelf stock material. Furthermore, the method considers the case where bars overlap at their joints (i.e. their thicknesses are added at the joint) and when they do not. Finally, our proposed method naturally accommodates the imposition of several fixed length scales. We demonstrate the proposed approach on classical problems of compliance-based topology optimization and identify its benefits as well as research directions to be addressed in the future.

2504 グレースケールフリートポロジー最適化のための高品質ゼロレベル境界追跡メッシュ作成法(設計と最適化IV)

Topology optimization is one approach of structural optimization, where various and complex shape can be represented. Because of its degree of freedom, topology optimization has potential to yield superior designs compared with sizing and shape optimizations. However, grayscale elements are often yielded, especially, in density based topology optimization methods; they influenc negatively on design problems where Dirichlet boundary conditions are imposed on the design boundary. To achieve grayscale-free topology optimization, we proposed a boundary tracking type level-set topology optimization method for three-dimensional problem in previous studies. However, the three-dimensional zero-level boundary tracking mesh often contains extremely-low quality elements, and these elements influenc negatively on finit element computation. To avoid this problem, we need a new algorithm to generate the high quality zero-level boundary tracking mesh. In this study, we propose a new mesh generation method for high quality zero-level boundary tracking mesh.

Filter the shape sensitivity in level set-based topology optimization methods

Filtering methods are widely used in density-based topology optimization methods, such as the SIMP method, to prevent checkerboards and mesh dependency due to their ease of implementation and their efficiency. In this study, several filtering schemes are presented to filter the shape sensitivity in level set-based structural topology optimization methods. It is revealed that filtering of the shape sensitivity can yield convergence with less iterations without considerably increasing the computing time of each iteration step. Thus, it can improve the overall computational efficiency. The validity of the method is tested on both the mean compliance minimization problem and the compliant mechanisms design problem.

Bi-directional evolutionary structural optimization for design-dependent fluid pressure loading problems

This article presents an evolutionary topology optimization method for compliance minimization of structures under design-dependent pressure loads. In traditional density based topology optimization methods, intermediate values of densities for the solid elements arise along the iterations. Extra boundary parametrization schemes are demanded when these methods are applied to pressure loading problems. An alternative methodology is suggested in this article for handling this type of load. With an extended bi-directional evolutionary structural optimization method associated with a partially coupled fluid–structure formulation, pressure loads are modelled with hydrostatic fluid finite elements. Due to the discrete nature of the method, the problem is solved without any need of pressure load surfaces parametrization. Furthermore, the introduction of a separate fluid domain allows the algorithm to model non-constant pressure fields with Laplace's equation. Three benchmark examples are explored in order to show the achievements of the proposed method.

1106 レベルセット法とゼロレベル境界追跡型メッシュによる3次元問題を対象としたグレースケールフリートポロジー最適化(OS5-2 設計と最適化2,OS5 設計と最適化)

Topology optimization is one approach of structural optimization, where various and complex shape can be represented. Because of its degree of freedom, topology optimization has potential to yield superior designs compared with sizing and shape optimizations. However, grayscale elements are often yielded, especially, in density based topology optimization methods; they influence negatively on design problems where Dirichlet boundary conditions are imposed on design boundary. To achieve grayscale-free topology optimization, we proposed to use the level-set method and the zero-level boundary tracking mesh in previous studies. The zero-level boundary tracking mesh is a conformed mesh to the zero iso-contour of the level-set function and it is generated from the Eulerian mesh. In our previous studies, only the two-dimensional problem were targeted. In this study, we propose a new mesh generation method for the three-dimensional zero-level boundary tracking mesh and construct a grayscale-free topology optimization method for three-dimensional problem.

Maximizing the fundamental eigenfrequency of geometrically nonlinear structures by topology optimization based on element connectivity parameterization

This paper pertains to the use of topology optimization based on the internal element connectivity parameterization (I-ECP) method for nonlinear dynamic problems. When standard density-based topology optimization methods are used for nonlinear dynamic problems, they typically suffer from two main numerical difficulties, element instability and localized vibration modes. As an alterative approach, the I-ECP method is employed to avoid element instability and a new patch mass model in the I-ECP formulation is developed to control the problem of localized vibration modes. After the I-ECP based formulation is developed, the advantages of the proposed method are checked with several numerical examples.

Adaptive volume constraint algorithm for stress limit-based topology optimization

Homogenization or density-based topology optimization methods work by distributing a fixed amount of material to the most effective areas of the design domain so as to create an optimal structural configuration that meets the minimum compliance criteria. These topology optimization methods generally cannot control the maximum stress levels of the structure; therefore, the smoothened optimum structure is not guaranteed to be ready for immediate use. This can be because it is either unsafe if the maximum stress at this structure exceeds the strength limit, or over designed if the maximum stress is far below the stress limit. Difficult and complex shape optimization must then be done to obtain a minimum-weight structure that meets the maximum stress constraint. This paper proposes an adaptive volume constraint (AVC) algorithm, a heuristic approach, in place of traditional topology optimization methods so that the maximum stress in the optimal structural configuration will be below the predefined stress limit and the smoothened structure will be directly applicable. In order to test the applicability and robustness of the AVC algorithm, topology optimization using both a traditional fixed volume constraint and an AVC are tested on a number of configuration design problems. To further illustrate the usefulness of the AVC algorithm, shape optimizations at the maximum stress constraint are also conducted on the smooth structural models by both optimization approaches on an identical problem set.