Multiresolution Topology Optimization Research Articles

Simultaneous discrete and continuum multiresolution topology optimization

Simultaneous discrete and continuum multiresolution topology optimization

Multi-resolution topology optimization using B-spline to represent the density field

This paper proposes a novel multi-resolution topology optimization method using B-spline to represent the density field, and overcomes the defects of tedious post-processing of element-based models and low computational efficiency of topology optimization for large-scale problems. The design domain embedded in the B-spline space is discretized with a coarser analysis mesh and a finer density mesh to reduce the computational cost of finite element analysis. As design variables, the coefficients of the control points control the shape of the B-spline. The optimized B-spline can be quickly and precisely converted into a CAD model. Sensitivity filtering is additionally applied to enhance B-spline's smoothness and suppress QR-patterns. Numerical examples, including 2D and 3D cases, are tested to demonstrate that the proposed method significantly saves computational time without sacrificing the performance of the optimized structure. Moreover, the post-processing procedures are streamlined, resulting in continuous, smooth, and editable models.

Polygonal multiresolution topology optimization of multi-material structures subjected to dynamic loads

Polygonal multiresolution topology optimization of multi-material structures subjected to dynamic loads

Multi-resolution Topology Optimization Using B-spline to Represent the Density Field

Multi-resolution Topology Optimization Using B-spline to Represent the Density Field

Density gradient‐based adaptive refinement of analysis mesh for efficient multiresolution topology optimization

AbstractIn topology optimization, the finite‐element analysis of the problem is generally the most computationally demanding task of the solution process. In order to improve the efficiency of this phase, in this article we propose to represent regions with zero density gradient by a coarser analysis mesh. The design is instead represented in a uniform mesh. We motivate the density gradient‐based adaptive refinement by discussing the topological meaning of the density gradient and how it can help avoid loss of information during projections or interpolations between design and analysis meshes. We also study the adaptiveness of the mesh and its ability to detect the topology change of the design. An a posteriori error analysis is performed as well. Furthermore, we provide theoretical and numerical considerations on the reduction of the number of degrees of freedoms of the adaptive analysis mesh with respect to the uniform case. This translates into a faster solution of the analysis, as we show numerically. Finally, we solve several test problems, including large 3D problems that we solve in parallel on computer cluster, demonstrating the applicability of our procedure in large scale computing and with iterative solvers.

Substructuring multi-resolution topology optimization with template

In the multi-resolution topology optimization (MTOP) method, by decoupling the finite element mesh and discretization of density field, the finite element analysis is carried out with a coarser mesh (i.e., super-elements), and the computational cost is thus greatly reduced in the process of topology optimization. However, the elemental stiffness matrix is calculated each iteration using the average density of super-elements, and this treatment is actually not only inaccurate but also leads to the checkerboard phenomenon and QR patterns when the filter radius is relatively small. In order to alleviate such issues, the super-element is treated as a substructure and the corresponding elemental stiffness matrix is obtained using static condensation. Furthermore, a template library is developed for the substructure based on its density distribution during the topology optimization process. By this means, the elemental stiffness matrix is not required to be calculated repeatedly, the accuracy of the MTOP method is improved significantly and the checkerboard patterns are effectively inhibited as well.

A three-dimensional multiscale approach to optimal design of porous structures using adaptive geometric components

This paper presents a direct multiscale design approach for three-dimensional (3D) porous structures. The porous material modeling technique, which involves the use of the adaptive geometric components (AGCs), is employed to simultaneously describe the macrostructure and microstructure of material while the multiresolution topology optimization (MTO) is utilized to reduce the cost of finite element analysis (FEA). We use two distinct discretization levels, including a coarse mesh to perform FEA (displacement mesh) and a fine mesh to present material distribution (density mesh). The AGCs are smoothly mapped onto the density mesh to obtain an effective element density field for material interpolation and stiffness computation of displacement-mesh elements. The overall macrostructure and microstructure of the material are simultaneously optimized by straightforwardly optimizing the geometry parameters of the AGCs. The 3D porous multiscale design with non-uniform microstructure can be directly achieved with only a global volume constraint. Some numerical tests are provided to validate the applicability of the proposed approach.

Lagrangian–Eulerian multidensity topology optimization with the material point method

AbstractIn this paper, a hybrid Lagrangian–Eulerian topology optimization (LETO) method is proposed to solve the elastic force equilibrium with the Material Point Method (MPM). LETO transfers density information from freely movable Lagrangian carrier particles to a fixed set of Eulerian quadrature points. This transfer is based on a smooth radial kernel involved in the compliance objective to avoid the artificial checkerboard pattern. The quadrature points act as MPM particles embedded in a lower‐resolution grid and enable a subcell multidensity resolution of intricate structures with a reduced computational cost. A quadrature‐level connectivity graph‐based method is adopted to avoid the artificial checkerboard issues commonly existing in multiresolution topology optimization methods. Numerical experiments are provided to demonstrate the efficacy of the proposed approach.

Multiresolution Topology Optimization of Large-Deformation Path-Generation Compliant Mechanisms with Stress Constraints

Topology optimization is a powerful numerical tool in the synthesis of lightweight structures and compliant mechanisms. Compliant mechanisms present challenges for topology optimization, as they typically exhibit large displacements and rotations. Path-generation mechanisms are a class of mechanisms that are designed to follow an exact path. The characteristics of compliant mechanisms therefore exclude the validity of linear finite-element analysis to ensure the proper modeling of deformation and stresses. As stresses can exceed the limit when neglected, stress constraints are needed in the synthesis of compliant mechanisms. Both nonlinear finite-element analysis as well as the consideration of stress constraints significantly increase computational cost of topology optimization. Multiresolution topology optimization, which employs different levels of discretization for the finite-element analysis and the representation of the material distribution, allows an important reduction of computational effort. A multiresolution topology optimization methodology is proposed integrating stress constraints based on nonlinear finite-element analysis for path-generation mechanisms. Two objective formulations are used to motivate and validate this methodology: maximum-displacement mechanisms and path-generation mechanisms. The formulation of the stress constraints and their sensitivities within nonlinear finite-element analysis and multiresolution topology optimization are explained. We introduce two academic benchmark examples to demonstrate the results for each of the objective formulations. To show the practical, large-scale application of this method, results for the compliant mechanism structure of a droop-nose morphing wing concept are shown.

A Two-Scale Multi-Resolution Topologically Optimized Multi-Material Design of 3D Printed Craniofacial Bone Implants.

Bone replacement implants for craniofacial reconstruction require to provide an adequate structural foundation to withstand the physiological loading. With recent advances in 3D printing technology in place of bone grafts using autologous tissues, patient-specific additively manufactured implants are being established as suitable alternates. Since the stress distribution of these structures is complicated, efficient design techniques, such as topology optimization, can deliver optimized designs with enhanced functionality. In this work, a two-scale topology optimization approach is proposed that provides multi-material designs for both macrostructures and microstructures. In the first stage, a multi-resolution topology optimization approach is used to produce multi-material designs with maximum stiffness. Then, a microstructure with a desired property supplants the solid domain. This is beneficial for bone implant design since, in addition to imparting the desired functional property to the design, it also introduces porosity. To show the efficacy of the technique, four different large craniofacial defects due to maxillectomy are considered, and their respective implant designs with multi-materials are shown. These designs show good potential in developing patient-specific optimized designs suitable for additive manufacturing.

Multi-resolution topology optimization using adaptive isosurface variable grouping (MTOP-aIVG) for enhanced computational efficiency

Because finite elements and density elements are separated in multi-resolution topology optimization (MTOP), a relatively fewer number of finite elements can be used, thereby significantly reducing computing cost in finite element analysis (FEA) during topology optimization. However, for large-scale problems, numerous design variables are still required to precisely represent the optimum topology. This causes a dominant computational burden in design optimization. In this paper, an efficient multi-resolution topology optimization (MTOP) using adaptive isosurface variable grouping (aIVG) is proposed to alleviate the above computational burden in topology optimization by grouping design variables of similar grouping criteria into a single grouped design variable. Adaptive isosurface variable grouping is performed according to the grouping criterion which can be calculated using design variables and their sensitivities. Numerical examples such as 2D and 3D compliance minimization, 2D compliant mechanism, 2D multiple displacement constraints, and 3D thermal compliance minimization demonstrate that the proposed MTOP-aIVG significantly reduces computation time in optimization by virtue of using a reduced number of design variables.

An efficient multi-resolution topology optimization scheme for stiffness maximization and stress minimization

This article develops a multi-resolution topology optimization (MTOP) approach based on the solid isotropic material with penalization (SIMP) method, which is effective in obtaining high-resolution designs at low computational cost. The extended finite element method (XFEM) is employed to decouple the analysis mesh, material description and nodal design variables. By the advantage of XFEM at modelling material discontinuity, detailed geometrical features are generated on a coarse analysis mesh. To obtain a clear interface between material grids, a variation of the traditional sensitivity filter is introduced to produce discrete solutions. The low computational costs make the proposed approach appropriate for dealing with problems requiring a high number of finite element analysis (FEA) processes, typically high-resolution/large-scale models, stress minimization, etc. Accurate von Mises stress is calculated on a high number of Gaussian points, making the approach perform better at stress minimization. Then, several 2D and 3D examples optimized by different solvers are illustrated to demonstrate the effectiveness and excellent generality of the proposed approach.

Image-Based Multiresolution Topology Optimization Using Deep Disjunctive Normal Shape Model

We present a machine learning framework for predicting the optimized structural topology designs using multiresolution data. Our approach primarily uses optimized designs from inexpensive coarse mesh finite element simulations for model training and generates high resolution images associated with simulation parameters that are not previously used. Our cost-efficient approach enables the designers to effectively search through possible candidate designs in situations where the design requirements rapidly change. The underlying neural network framework is based on a deep disjunctive normal shape model (DDNSM) which learns the mapping between the simulation parameters and segments of multi resolution images. Using this image-based analysis we provide a practical algorithm which enhances the predictability of the learning machine by determining a limited number of important parametric samples (i.e. samples of the simulation parameters) on which the high resolution training data is generated. We demonstrate our approach on benchmark compliance minimization problems including the 3D topology optimization where we show that the high-fidelity designs from the learning machine are close to optimal designs and can be used as effective initial guesses for the large-scale optimization problem.

Three-dimensional high resolution topology optimization considering additive manufacturing constraints

This paper studies additive manufacturing oriented structural topology optimization with SIMP approach and aims at 3D high-resolution printable structural topology design with overhang and horizontal minimum length control for minimum compliance.To start with, we construct a hyperplane in ℝ4 by fitting a local element density distribution in the 18 Elements Scheme, use its gradient to estimate the overhang angle, the directional-dependent overhang angle and formulate the corresponding constraints. Next, we propose a Horizontal Square Scheme and four support sets around the concerned element. The horizontal minimum length is controlled by forbidding the concerned element’s density to be larger than the average density of the elements in one of the support sets. By combining these two constraints, the hanging feature is well suppressed.A new implementation scheme with an improved weight function is proposed to meet these element wise AM constraints well. To get high resolution structural boundaries with low computational efforts, this paper applies the multiresolution topology optimization (MTOP) method.The structural TO problem is solved by MMA. A number of numerical examples and AM experiments show the effectiveness of this method. The present approach works efficiently when the building direction is in slight misalignment with the vertical direction.

An efficient evolutionary structural optimization method for multi-resolution designs

This paper presents a computationally efficient multi-resolution topology optimization framework by establishing a novel bi-directional evolutionary structure optimization (BESO) method based on extended finite element method (XFEM). In the proposed framework, the high-resolution designs preserving the topological complexity can be obtained with low degree of freedoms (DOFs). The implementation of the presented multi-resolution optimization framework takes good use of the ability of XFEM at accurately modeling material discontinuities within one element. On the basis of XFEM, a strategy of triangulated partition and a new material interpolation model are introduced to represent the finer material distribution. We employ the coarser finite element (FE) mesh to perform the finite element analysis, the sub-parts partitioned from finite elements to describe material properties and the nodal design variables to perform the optimization. To circumvent artificially stiff patterns, a modified sensitivity filter is applied to regularize the solution. The effectiveness and high efficiency of the presented approach are highlighted by typical 2D and 3D examples.

Design and analysis adaptivity in multiresolution topology optimization

SummaryMultiresolution topology optimization (MTO) methods involve decoupling of the design and analysis discretizations, such that a high‐resolution design can be obtained at relatively low analysis costs. Recent studies have shown that the MTO method can be approximately 3 and 30 times faster than the traditional topology optimization method for two‐dimensional (2D) and three‐dimensional (3D) problems, respectively. To further exploit the potential of decoupling analysis and design, we propose a dp‐adaptive MTO method, which involves locally increasing/decreasing the polynomial degree of the shape functions (p) and the design resolution (d). The adaptive refinement/coarsening is performed using a composite refinement indicator that includes criteria based on analysis error, presence of intermediate densities, as well as the occurrence of design artifacts referred to as QR‐patterns. While standard MTO must rely on filtering to suppress QR‐patterns, the proposed adaptive method ensures efficiently that these artifacts are suppressed in the final design, without sacrificing the design resolution. The applicability of the dp‐adaptive MTO method is demonstrated on several 2D mechanical design problems. For all the cases, significant speedups in computational time are obtained. In particular for design problems involving low material volume fractions, speedups of up to a factor of 10 can be obtained over the conventional MTO method.

An isogeometric approach to topology optimization of spatially graded hierarchical structures

This paper presents an isogeometric approach to topology optimization of spatially graded hierarchical structures, which are assumed to be consisted of identical or spatially varying substructures. In this work, the isogeometric analysis (IGA) is adopted for an efficient and accurate performance assessment of spatially graded structures, especially for those with complex geometries. A multi-resolution topology optimization (MTOP) scheme with two distinct mesh discretization levels is used to achieve high-resolution topology representation. In specific, displacement field and topology field are approximated separately on two distinct levels of mesh discretization, where the IGA is performed on a coarse mesh and nodal-based topology design variables are defined on a refined mesh obtained by k-refinement scheme. A projection scheme with the use of non-uniform rational basis spline (NURBS) basis functions is employed to compute the densities at quadrature points from nodal densities variables. The proposed IGA-MTOP-based design framework has threefold merits: a) global structure and constituent substructures are strictly coupled with no assumption on the separation of scales as required in homogenization-based designs, which together with high continuous NURBS basis functions guarantee automatically smooth connection between substructures; b) high-resolution designs with detailed local structural features can be obtained in a computationally high-efficient manner; c) the substructures designed in the common parameter space can be freely transformed to conform the physical space. The effectiveness and robustness of the proposed method are validated by a series of benchmark designs.

An efficient moving morphable component (MMC)-based approach for multi-resolution topology optimization

In the present work, a highly efficient Moving Morphable Component (MMC) based approach for multi-resolution topology optimization is proposed. In this approach, high-resolution optimization results can be obtained with much less number of degrees of freedoms (DOFs) and design variables since the finite element analysis model and the design optimization model are totally decoupled in the MMC-based problem formulation. This is achieved by introducing super-elements for structural response analysis and adopting a domain decomposition strategy to preserve the topology complexity of optimized structures. Both two-and three-dimensional numerical results demonstrate that substantial computational efforts can be saved with use of the proposed approach.

Conceptual design of efficient heat conductors using multi-material topology optimization

ABSTRACTConductive heat transfer plays an important role in dissipating thermal energy to achieve lower operating temperatures in various devices. Topology optimization has the potential to provide efficient structural solutions for such devices. The traditional topology optimization approach considers a single material. Adding additional materials with unique properties not only can expand the design options but also may improve the structural performance of the final structure. In this work, a multi-resolution topology optimization approach is employed to design multi-material structures for efficient heat dissipation. The implementation blends an efficient multi-resolution approach to obtain high-resolution designs with an alternating active phase algorithm to handle multi-material giving greater design flexibility. It solves the steady-state heat equation using finite element analysis and iteratively minimizes thermal compliance (maximizes conductivity). Several examples are presented to show the efficacy of the numerical implementation, which involves benchmark problems. Results indicate good prospects when quantitatively compared with single-material structures.

QR-patterns: artefacts in multiresolution topology optimization

Recent multiresolution topology optimization (MTO) approaches involve dividing finite elements into several density cells (voxels), thereby allowing a finer design description compared to a traditional FE-mesh-based design field. However, such formulations can generate discontinuous intra-element material distributions resembling QR-patterns. The stiffness of these disconnected features is highly overestimated, depending on the polynomial order of the employed FE shape functions. Although this phenomenon has been observed before, to be able to use MTO at its full potential, it is important that the occurrence of QR-patterns is understood. This paper investigates the formation and properties of these QR-patterns, and provides the groundwork for the definition of effective countermeasures. We study in detail the fact that the continuous shape functions used in MTO are incapable of modeling the discontinuous displacement fields needed to describe the separation of disconnected material patches within elements. Stiffness overestimation reduces with p-refinement, but this also increases the computational cost. We also study the influence of filtering on the formation of QR-patterns and present a low-cost method to determine a minimum filter radius to avoid these artefacts.