Multimaterial topology optimization for finite strain elastoplasticity: Theory, methods, and applications

<div class="section abstract"><div class="htmlview paragraph">Evolving fuel efficiency and emissions standards, along with consumer demand for performance, are strong pressures for light-weighting of performance oriented motorcycles. The field of topology optimization (TO), with the extension of multi-material topology optimization (MMTO) provide manufacturers with advanced structural light-weighting methodology. TO methodology has been adopted in many industries, including automotive where light-weighting assists in meeting efficiency regulations. The development of process specific manufacturing constraints within MMTO is a critical step in increasing adoption within industries dealing with manufacturing cost restrictions. This capability can decrease design complexity, lowering manufacturing costs of optimization solutions.</div><div class="htmlview paragraph">A conventional all-aluminum perimeter style motorcycle chassis is analyzed to develop baseline compliance (total strain energy) metrics. An MMTO design space is created and optimized with steel and aluminum, such that results match the baseline design weight. This formulation demonstrates increased structural efficiency through stiffer structures at equivalent weight. Results are generated with standard MMTO, symmetry, and extrusion constraints to demonstrate utility of these manufacturing constraints. Material ratios are used to enforce lower cost material distribution selections.</div><div class="htmlview paragraph">The usage of MMTO with manufacturing and material ratio constraints has resulted in up to 60.4% reduction in structural compliance of designs, and a multitude of lower cost alternative designs. The usage of symmetry constraints provides effectively identical results to standard MMTO, with a computational time reduction of 29%. Extrusion constraints demonstrate decreased manufacturing difficulty with a computational time reduction of 52% and structural performance penalty of 58.6% from the lowest compliance MMTO result. The enforcement of steel has demonstrated a decrease in structural performance (increase in compliance) and material costs, with varying degrees depending on manufacturing constraints and enforcement limits. The MMTO based designs provide a range of solutions to designers, which can be selected based on the importance of structural efficiency, manufacturing difficulty, and material costs.</div></div>

ABSTRACTA comprehensive solution for bus frame design is proposed to bridge multi-material topology optimization and cross-sectional size optimization. Three types of variables (material, topology and size) and two types of constraints (static stiffness and frequencies) are considered to promote this practical design. For multi-material topology optimization, an ordered solid isotropic material with penalization interpolation is used to transform the multi-material selection problem into a pure topology optimization problem, without introducing new design variables. Then, based on the previously optimal topology result, cross-sectional sizes of the bus frame are optimized to further seek the least mass. Sequential linear programming is preferred to solve the two structural optimization problems. Finally, an engineering example verifies the effectiveness of the presented method, which bridges the gap between topology optimization and size optimization, and achieves a more lightweight bus frame than traditional single-material topology optimization.

The algorithms for multi-material topology optimization were developed to solve compliance-minimization problems and applied to engineering problems in automotive concepts and lightweight design. Two small-scale problems of a long cantilever and a control arm were studied initially to verify the effectiveness of the developed algorithms and in-house program. Optimal solutions achieved by the multi-material topology optimization method developed were compared to their counterparts obtained by standard single-material topology optimization. To efficiently solve real-world engineering problems, the algorithms were further advanced to incorporate extrusion constraints and to handle multiple load cases. The effectiveness and the efficiency of the proposed method were demonstrated by the study of two real-world engineering problems: (a) the conceptual design of a cross-member for a chassis frame; and (b) the conceptual design of an automotive engine cradle. The two optimization design problems both involved complex geometries, design and non-design domains, prescribed regions with specific material allocations, multiple load cases, and manufacturing extrusion constraints. It was explicitly demonstrated that, for the same weight, the optimum designs achieved by the multi-material topology optimization method were stiffer than those achieved by standard single-material topology optimization.

Generalized standard media and variational principles in classical and finite strain elastoplasticity

This paper presents a new strategy to distribute two different materials for multi-material topology optimization. Extended from the bi-directional evolutionary structural optimization (BESO) method for a single material, the multi-material bidirectional evolutionary structural optimization (MBESO) method has been developed, which can effectively handle the topology optimization problems involving two materials like steel and concrete. However, in some special cases, overloading of part of the compressed material occurs in the multi-material structures designed by using the MBESO method. Aimed to solve this critical problem, a simple but effective strategy is proposed in this paper. In steel-concrete composite structures, for instance, the overloaded compressed concrete elements with exceedingly high stress are replaced with steel material. The small amount of steel material added to the highly compressed region can effectively reduce the maximum compressive stress of the concrete material to a safe level. The comparison between the original MBESO method and the improved strategy based on a series of two-dimensional and three-dimensional examples clearly demonstrates the effectiveness of the proposed strategy in enhancing the structural safety and strength of the topologically optimized composite structures. This distinctly different material distribution strategy shows its potential and value in multi-material topology optimization research and applications.

PurposeThis paper aims to present a deep learning–based surrogate model for fast multi-material topology optimization of an interior permanent magnet (IPM) motor. The multi-material topology optimization based on genetic algorithm needs large computational burden because of execution of finite element (FE) analysis for many times. To overcome this difficulty, a convolutional neural network (CNN) is adopted to predict the motor performance from the cross-sectional motor image and reduce the number of FE analysis.Design/methodology/approachTo predict the average torque of an IPM motor, CNN is used as a surrogate model. From the input cross-sectional motor image, CNN infers dq-inductance and magnet flux to compute the average torque. It is shown that the average torque for any current phase angle can be predicted by this approach, which allows the maximization of the average torque by changing the current phase angle. The individuals in the multi-material topology optimization are evaluated by the trained CNN, and the limited individuals with higher potentials are evaluated by finite element method.FindingsIt is shown that the proposed method doubles the computing speed of the multi-material topology optimization without loss of search ability. In addition, the optimized motor obtained by the proposed method followed by simplification for manufacturing is shown to have higher average torque than a reference model.Originality/valueThis paper proposes a novel method based on deep learning for fast multi-material topology optimization considering the current phase angle.

Multi-material topology optimization of laminated composite beam cross sections

Microstructure design of porous viscoelastic composites with prescribed relaxation moduli by multi-material topology optimization

We discover that the given initial structural density value has significant influence on the final result of multi-material topology optimization when adopting the SIMP interpolation scheme. The influence cannot be ignored if the grid is sparse, and there is a certain degree of periodic relationship between the structural minimum compliance and the given initial density value. By modifying the initial structural density value, a better topology optimization result with a smaller compliance can be obtained. The expression is proposed on how to modify the initial structural density value to get better results in multi-material topology optimization problems.

Adaptive isogeometric multi-material topology optimization based on suitably graded truncated hierarchical B-spline

This paper presents a multi-material topology optimization for the design of permanent magnet synchronous motors (PMSMs). Specifically, structural shapes of permanent magnet (PM) and iron core in a PMSM rotor are simultaneously designed together with the orientation of PM magnetization. For a co-design of PM and iron core, relative permeability and residual magnetic flux density are interpolated by the three-field density approach based on the Helmholtz filtering and regularized Heaviside step function. Here, the Helmholtz filtering aims to attain smooth border in design results, and the Heaviside function enables us to acquire a clear border (i.e. zero-one design) without intermediate densities. The optimization goal is set as maximizing the average torque of PMSMs. The average torque is calculated by Maxwell stress tensor (MST) method considering a maximum torque per ampere (MTPA) control. To validate the effectiveness of the proposed multi-material topology optimization approach, a PMSM rotor with 4 poles and 12 slots is designed. In addition, design results at various settings of input current amplitude and PM strength are compared and discussed. When the input current is stronger than the PM strength, rotor PM and iron core are designed for utilizing both PM and reluctance torque components like V-shape interior PMSMs. On the other hand, in the case of stronger PM strength, PM is designed near the air-gap, which utilizes only PM torque component like surface PMSMs.

A heuristic approach to handle strength constraints based on material failure criteria in multi-material topology optimization is presented. This is particularly advantageous if different materials have different failure criteria. The change in the material failure function in an element due to a contemplated material change is estimated without the need for expensive matrix factorizations. This change is used along with the changes to the objective and deflection-based constraint functions, computed using pseudo-sensitivities, to determine a single aggregated ranking parameter for the element. Elements are ranked on the basis of their ranking parameters and this rank is used to modify the material ID-s of a few top-ranked elements during an optimization iteration. The working of the algorithm is demonstrated on a few example problems showing its effectiveness and utility in deriving optimal topologies with multiple materials in the presence of stress and strain-based failure criteria, in addition to the conventional stiffness-based constraints.

Multi Material Topology Optimization with Hybrid Cellular Automata

Topology optimization has been considered as a promising tool for conceptual design due to its capability of generating innovative design candidates without depending on the designer's intuition and experience. Various optimization methods have been developed through the years, and one of the promising options is the level-set-based topology optimization method. The benefit of this alternative method is that the design is characterized by its clear boundaries. This advantage can avoid postprocessing work in conventional topology optimization process to a large extent and realize direct integration between topology optimization and additive manufacturing (AM). In this paper, practical algorithms and a matlab-based open source framework are developed to seamlessly integrate the level-set-based topology optimization procedure with AM process by converting the design to STereoLithography (STL) files, which is the de facto standard format for three-dimensional (3D) printing. The proposed algorithm and code are evaluated by a proof-of-concept demonstration with 3D printing of both single and multimaterial topology optimization results. The algorithm and the open source framework proposed in this paper will be beneficial to the areas of computational design and AM.

Topology optimization with a finite strain nonlocal damage model using the continuous adjoint method