Efficient Topology Optimization Method Research Articles

Fail-Safe Topology Optimization Using Damage Scenario Filtering

Within the framework of isotropic materials, this paper introduces an efficient topology optimization method that incorporates fail-safe design considerations using a penalty function approach. Existing methods are either computationally expensive or overlook fail-safe requirements during optimization. This approach not only achieves optimized structures with fail-safe characteristics, but also significantly enhances the computational efficiency of fail-safe topology optimization. In this method, the minimization of worst-case compliance serves as the optimization objective, employing the Kreisselmeier–stein Hauser function to approximate the non-differentiable maximum operator. A sensitivity analysis, derived through the adjoint method, is utilized, and a universal fail-safe optimization criterion is developed to update the design variables. During the optimization process for fail-safe strategies, a density-based filtering method is applied, effectively reducing damage scenarios. Finally, the effectiveness and computational efficiency of this method are validated through several numerical examples.

A topology optimization framework for 3D phononic crystals via the method of successive iteration of analysis and design

Topology optimization has become a useful tool for designing Phononic Crystals (PnCs) to achieve desired elastic wave propagation properties. However, topological design of three-dimensional (3D) PnCs remains a significant challenge due to extremely high computational cost required for the repeated solution of the nested double-loop problem of design optimization and eigenvalue analysis to determine the band structure during the optimization iteration process. This paper presents a more efficient topology optimization method based on the concept of successive iteration of analysis and design. The method employs a sequential approximation of the eigenpairs through an inverse iteration-like procedure to avoid solving the expensive eigenvalue problems in each design iteration. To further improve the efficiency of the design iteration, the eigenmodes of the intermediate PnCs designs are first computed on a relatively coarse finite element mesh. Then they are mapped onto the fine mesh and used as the initial trial eigenmodes in the eigenpair analysis and design sensitivity analysis. The high efficiency of the proposed method, as compared with conventional nested double-loop optimization approaches, is demonstrated by 3D numerical examples with over one million degrees of freedom. It is shown that this approach can achieve simultaneous convergence of the eigenvectors along with the design evolution of the unit cell, therefore substantially reduces the computational burden for high-resolution topological design of 3D PnCs.

Three-dimensional metamaterials exhibiting extreme isotropy and negative Poisson's ratio

This research will develop a new set of mechanical metamaterials with simultaneous ideal elastic isotropy and extreme negative Poisson's ratio through a topology optimization approach. Firstly, we propose a new method to generate a three-dimensional ground structure with geometric cubic symmetry. Secondly, a more efficient topology optimization method is developed by utilizing a small number of both integer and continuous design variables for topological description of the design based on the discrete ground structure. Thirdly, two new geometric constraints are incorporated into the mathematical formulation to ensure creation of designs that are practical. Finally, the genetic algorithm is used to solve this type of mixed-integer programming problems. Several numerical examples have been used to demonstrate the generation of a series of three-dimensional microarchitectures from a single integrated design. The microlattices can also be customized to obtain a broad range of negative Poisson's ratios across different relative densities. The effective material properties and macroscopic deformation behaviors of the designs are numerically validated using commercial software.

An efficient topology optimization method based on adaptive reanalysis with projection reduction

An efficient topology optimization based on the adaptive auxiliary reduced model reanalysis (AARMR) method is proposed to improve computational efficiency and scale. In this method, a projection auxiliary reduced model (PARM) is integrated into the combined approximation reduced model (CARM) to reduce the dimension of the model in different aspects. First, the CARM restricts the solution space to avoid large matrix factorization. Second, the PARM is proposed to construct the CARM dynamically to save computational cost. Furthermore, the multi-grid conjugate gradient method is suggested to update PARM adaptively. Finally, several classic numerical examples are tested to show that the proposed method not only significantly improves computational efficiency, but also can solve large-scale problems that are difficult to solve by direct solvers due to the memory limitations.

Topology optimization for structures with bi-modulus material properties considering displacement constraints

In this paper, an efficient topology optimization method is developed for bi-modulus structures considering displacement constraints. In order to overcome the numerical difficulty caused by the non-smoothness bi-modulus material properties, a new smooth elastic modulus matrix concerned with the principal strain states is firstly constructed to approximate the discontinuous modulus matrix for bi-modulus materials. Based on the smooth approximation, topology optimization model for structures under both single and multiple load cases is built to minimize structural compliance with volume and displacement constraints. Considering the nonlinear equilibrium equations, the sensitivities of the objective function and constraints are explicitly achieved based on the adjoint method, and then the gradient-based method of moving asymptotes algorithm is employed to update the topological design variables. A cooperation platform is constructed for the proposed bi-modulus topology optimization, which provides an effective tool for the design optimizations. The proposed method is proved to have good efficiency and applicability to deal with problems with multiple load cases and multiple constraints by several typical numerical examples.

Topology optimization design of deformable flexible thermoelectric devices for voltage enhancement

Flexible thermoelectric devices can generate electricity based on temperature differences and have great potential as an energy source for wearable devices and the Internet of Things. In this paper, an efficient topology optimization method is proposed for the design of deformable flexible thermoelectric devices, which facilitates an increase in the output voltage and a corresponding improvement in the power generation efficiency. The designed structures can produce out-of-plane displacement for an external load, which increases the temperature difference between the hot and cold ends, thus enhancing the power generation capacity. Using combined multi-field coupling analysis and experiments, the efficacy of the proposed method was evaluated for several scenarios based on different load cases and design domains. After optimization, the temperature difference of thermoelectric devices can be increased by more than 10 °C, and the output voltage can exhibit a three-fold increase compared to that of the in-plane configuration.

Thermal-fluid topology optimization with unconditional energy stability and second-order accuracy via phase-field model

This paper aims to establish a novel and efficient topology optimization method for the thermal-fluid. To adaptively design the fluid–solid coupling structure and make the objective energy to dissipate, the proposed method considers several constraints, such as the volume conservation, inlet and outlet flow velocity field and fluid–solid boundary constraints. The governing system includes the phase-field model, the steady state Darcy equation and the heat transfer equation. Under the constraints of multiple physical fields, we prove the existence of minimal solutions to the optimization problem. We use a Crank–Nicolson (CN) type scheme to discretize the governing system. The multigrid method is used to solve the resulting system of discrete equations. We prove the boundedness and unconditional stability of the original energy, which implies that a large time step can be used. The proposed discrete system is both spatially and temporally second-order accurate. Various computational tests have been performed to demonstrate that the numerical approach is efficient in designing the complicated structures of thermal fluid flows.

Cube-Tet transformation method accelerating the process of topology optimization

ABSTRACT High-resolution topology optimization ensures highly accurate solutions and structural details; however, it also imposes a heavy computational burden. In this study, an efficient topology optimization method based on Cube-Tet mesh is proposed that significantly shortens the duration of finite element analysis (FEA) by compressing the size of the sparse matrix. First, the uniform cubic voxel set is obtained from the model. Each cubic voxel is transformed into six tetrahedral voxels without changing the total number of nodes; then, the FEA is performed. By transforming the mesh style, the coupling terms between nodes are decreased, the difficulty in solving the sparse matrix is reduced, and the iterative solution time is shortened, thus increasing the efficiency of topology optimization.

A Transitional Connection Method for the Design of Functionally Graded Cellular Materials

In recent years, the functionally graded materials (FGM) with cellular structure have become a hot spot in the field of materials research. For the continuously varying cellular structure in the layer-wise FGM, the connection of gradient cellular structures has become the main problem. Unfortunately, the effect of gradient connection method on the overall structural performance lacks attention, and the boundary mismatch has enormous implications. Using the homogenization theory and the level set method, this article presents an efficient topology optimization method to solve the connection issue. Firstly, a simple but efficient hybrid level set scheme is developed to generate a new level set surface that has the partial features of two candidate level sets. Then, when the new level set surface is formed by considering the level set functions of two gradient base cells, a special transitional cell can be constructed by finding the zero level set of this generated level set surface. Since the transitional cell has the geometric features of two gradient base cells, the shape of the transitional cell fits perfectly with its connected gradient cells on both sides. Thus, the design of FGM can have a smooth connectivity with C1 continuity without any complex numerical treatments during the optimization. A number of examples on both 2D and 3D are provided to demonstrate the characteristics of the proposed method. Finite element simulation has also been employed to calculate the mechanical properties of the designs. The simulation results show that the FGM devised by the proposed method exhibits better mechanical performances than conventional FGM with only C0 continuity.

An Enhanced Topology Optimization Approach Based on the Combined MMC and NURBS-Curve Boundaries

An efficient topology optimization method is developed newly in this study by combining the Non-uniform rational B-spline (NURBS) curves and moving morphable components (MMC). The MMC-based topology optimization is an explicit and geometrical method that utilizes a set of morphable components to create basic blocks for optimization. Optimum topologies may be obtained by optimizing shapes, lengths, thicknesses, orientations and layout of these components. The combined method adopts a different way in the creation of morphable components that consist of NURBS curves. Various kinds of complicated curved components can be built with NURBS curves or surfaces. Here, the NURBS curve is applied for shaping the geometries of structural basic components, and the coordinates of control points become design variables for topology optimization. A MATLAB optimization code has been developed. Four numerical examples of a short cantilever, a MBB beam, a simply supported beam with two point loadings, and a vehicle lower chassis structure subjected to crash loadings are provided to prove that the combined topology optimization approach coupled with NURBS curves and basic morphable components can get optimum topologies with clear topological boundaries successfully. As results of comparison study with other approaches, we can obtain the same topologies and faster convergence rates for the three separate cases. The combined approach can improve the smoothness of the topological boundaries that are similar to the shape optimization results obtained by post-optimization after the density-based topology optimization.

Topology optimization for removing internal resonances of a rotating thin plate

Internal resonance allows energy to transfer between eigenmodes and sometimes results in undesirable phenomena. Topology optimization has the ability to control the eigenfrequencies of a structure and is therefore possible to remove the internal resonances. In this study, topology optimization is studied for removing internal resonances of a rotating rectangular thin plate. First of all, an accurate dynamic model is established for the rotating thin plate by considering the Coriolis force. The thin plate element of the absolute nodal coordinate formulation (ANCF) is used to discretize the rotating thin plate and establish the dynamic equation. When the dynamic equation is transformed from the absolute frame to the rotating frame for modal analysis, the Coriolis force term shows up and is analytically derived. Then, an explicit and efficient topology optimization method, i.e., the method of moving morphable components (MMC) is employed to optimize the rotating thin plate. In order to remove the internal resonances, the detuning parameters are defined so that removing the internal resonances is equivalent to maximizing the minimum absolute detuning parameter. The corresponding max-min optimization problem can be dealt with via a bound formulation. Finally, three numerical examples are presented to validate the dynamic model and the topology optimization methodology. Two objectives are considered for optimization, i.e., removing the internal resonances of the rotating thin plate at different specific angular speeds and removing the most common 1:1 or 1:2 internal resonances of the rotating thin plate during a certain range of angular speeds.

Topology optimization of vibroacoustic problems using the hybrid finite element–wave based method

Determining an optimized solution by means of topology optimization in vibroacoustic problems often requires a high computational cost. Conventional density-based topology optimization using the finite element method is a time-consuming approach, owing to the large model size and repeated function evaluations involved in the frequency response. To address this issue, an efficient topology optimization method that uses the hybrid finite element–wave based method is proposed in this paper. In this method, the entire problem domain is divided into design and non-design domains. The mixed displacement–pressure finite element method is applied to the design domain for material interpolation in the topology optimization. Moreover, the wave based method, which is an efficient numerical scheme for acoustic problems, is applied to the non-design domain to reduce the computational cost. A direct coupling approach is proposed to construct hybrid models for vibroacoustic problems. The adjoint variable method is also presented to compute the design sensitivities efficiently. The effectiveness of the proposed method is demonstrated by benchmark problems. The optimization results indicate that the proposed method can significantly reduce the computational cost, while maintaining almost identical optimized layouts to those obtained using the conventional approach.

Efficient structure crash topology optimization strategy using a model order reduction method combined with equivalent static loads

In this study, an efficient topology optimization method under crash loads is proposed by combining the equivalent static loads with a model order reduction method, which is referred as the reduced model–based equivalent static loads method for nonlinear dynamic response topology optimization method. Considering that some parts of the vehicle experience large nonlinear deformations, whereas others exhibit only small linear deformations in a vehicle crash scenario, the linear and nonlinear behavior parts are identified and the whole model of the complete structure is divided into nonlinear and linear sub-models. At each cycle, the model order reduction method is used in the linear sub-model during crash analysis to solve the low-density-elements-induced mesh distortion problem and accelerate this process. In the linear static topology optimization, the nonlinear sub-model that was initially used to describe the nonlinear behavior part is linearized by the equivalent static loads method and then reduced by the Guyan reduction method. Then, the reduced equivalent static load model is assembled into the linear sub-model that is defined as the design space to formulate a reduced topology optimization model of the complete structure and the reduced equivalent static loads that only act on master degrees of freedom are calculated. Finally, the linear static topology optimization is performed based on the reduced topology optimization model with the reduced equivalent static loads to enhance the efficiency and improve the numerical stability. The process is repeated until the convergence criterion is satisfied. The effectiveness of the proposed method is demonstrated by investing a numerical example. The results show that the proposed method provides a feasible strategy for the topology optimization under crash loads, which can effectively improve the numerical stability and convergence.

Isogeometric topology optimization for continuum structures using density distribution function

SummaryThis paper will propose a more effective and efficient topology optimization method based on isogeometric analysis, termed as isogeometric topology optimization (ITO), for continuum structures using an enhanced density distribution function (DDF). The construction of the DDF involves two steps. (1) Smoothness: the Shepard function is firstly utilized to improve the overall smoothness of nodal densities. Each nodal density is assigned to a control point of the geometry. (2) Continuity: the high‐order NURBS basis functions are linearly combined with the smoothed nodal densities to construct the DDF for the design domain. The nonnegativity, partition of unity, and restricted bounds [0, 1] of both the Shepard function and NURBS basis functions can guarantee the physical meaning of material densities in the design. A topology optimization formulation to minimize the structural mean compliance is developed based on the DDF and isogeometric analysis to solve structural responses. An integration of the geometry parameterization and numerical analysis can offer the unique benefits for the optimization. Several 2D and 3D numerical examples are performed to demonstrate the effectiveness and efficiency of the proposed ITO method, and the optimized 3D designs are prototyped using the Selective Laser Sintering technique.

Topology optimization for auxetic metamaterials based on isogeometric analysis

In this paper, an effective and efficient topology optimization method, termed as Isogeometric Topology Optimization (ITO), is proposed for systematic design of both 2D and 3D auxetic metamaterials based on isogeometric analysis (IGA). Firstly, a density distribution function (DDF) with the desired smoothness and continuity, to represent the topological changes of structures, is constructed using the Shepard function and non-uniform rational B-splines (NURBS) basis functions. Secondly, an energy-based homogenization method (EBHM) to evaluate material effective properties is numerically implemented by IGA, with the imposing of the periodic boundary formulation on material microstructure. Thirdly, a topology optimization formulation for 2D and 3D auxetic metamaterials is developed based on the DDF, where the objective function is defined as a combination of the homogenized elastic tensor and the IGA is applied to solve the structural responses. A relaxed optimality criteria (OC) method is used to update the design variables, due to the non-monotonic property of the problem. Finally, several numerical examples are used to demonstrate the effectiveness and efficiency of the proposed method. A series of auxetic microstructures with different deformation mechanisms (e.g. the re-entrant and chiral) can be obtained. The auxetic behavior of material microstructures are numerically validated using ANSYS, and the optimized designs are prototyped using the Selective Laser Sintering (SLS) technique.

Topology Optimization of Periodic Structures With Substructuring

Topology optimization of macroperiodic structures is traditionally realized by imposing periodic constraints on the global structure, which needs to solve a fully linear system. Therefore, it usually requires a huge computational cost and massive storage requirements with the mesh refinement. This paper presents an efficient topology optimization method for periodic structures with substructuring such that a condensed linear system is to be solved. The macrostructure is identically partitioned into a number of scale-related substructures represented by the zero contour of a level set function (LSF). Only a representative substructure is optimized for the global periodic structures. To accelerate the finite element analysis (FEA) procedure of the periodic structures, static condensation is adopted for repeated common substructures. The macrostructure with reduced number of degree of freedoms (DOFs) is obtained by assembling all the condensed substructures together. Solving a fully linear system is divided into solving a condensed linear system and parallel recovery of substructural displacement fields. The design efficiency is therefore significantly improved. With this proposed method, people can design scale-related periodic structures with a sufficiently large number of unit cells. The structural performance at a specified scale can also be calculated without any approximations. What’s more, perfect connectivity between different optimized unit cells is guaranteed. Topology optimization of periodic, layerwise periodic, and graded layerwise periodic structures are investigated to verify the efficiency and effectiveness of the presented method.

An Efficient Topology Optimization Method for Structures with Uniform Stress

This paper focuses on two kinds of bi-objective topology optimization problems with uniform-stress constraints: compliance-volume minimization and local frequency response–volume minimization problems. An adaptive volume constraint (AVC) algorithm based on an improved bisection method is proposed. Using this algorithm, the bi-objective uniform-stress-constrained topology optimization problem is transformed into a single-objective topology optimization problem and a volume-decision problem. The parametric level set method based on the compactly supported radial basis functions is employed to solve the single-objective problem, in which a self-organized acceleration scheme based on shape derivative and topological sensitivity is proposed to adaptively adjust the derivative of the objective function and the step length during the optimization. To solve the volume-decision problem, an improved bisection method is proposed. Numerical examples are tested to illustrate the feasibility and effectiveness of the self-organized acceleration scheme and the AVC algorithm based on the improved bisection method. An extended application to the bi-objective stress-constrained topology optimization of a structure with stress concentration is also presented.

Topology Optimization of Piezoelectric Energy Harvesting Skin Using Hybrid Cellular Automata

An earlier study introduced the concept of piezoelectric energy-harvesting skin (EHS) to harvest energy by attaching thin piezoelectric patches onto a vibrating skin. This paper presents a methodology for the optimum design of EHS with the use of an efficient topology optimization method referred to as the hybrid cellular automaton (HCA) algorithm. The design domain of the piezoelectric material is discretized into cellular automata (CA), and the response of each CA is measured using high-fidelity finite-element analysis of a vibrating structure. The CA properties are parameterized using nonlinear interpolation functions that follow the principles of the SIMP model. The HCA algorithm finds the optimal densities and polarizing directions at each CA that maximize the output power from the EHS. The performance of this approach is demonstrated for the optimal design of EHS in two real-world case studies.

Topology optimization of gas flow channel routes in an automotive fuel cell

An efficient topology optimization method for fluid-structure problems was developed in an effort to determine the optimum flow channel route in a fuel cell bipolar plate from first principles. This study describes the derivation and solution of new mathematical equations for topology optimization combining a density-based algorithm, the interpolation method of moving asymptotes (MMA), and the incompressible Navier-Stokes equation with a term representing the chemical reaction between hydrogen and the catalyst. The present method is based on the finite element method with a newly developed reaction rate equation. In this model, a topology variable of 0 represents viscous flow, whereas a value of 1 indicates porous flow. The flow velocity and pressure were obtained from the Navier-Stokes equation and constraints and element matrices for sensitivity analyses during the optimization. MMA was utilized to calculate the optimum flow routes in the design domain. The influence of the key design parameter q and the pressure drop on the optimum topology were also investigated. The channel topology became smoother with decreasing q, and the number of channels increased with increasing pressure drop.