Morphology-based black and white filters for topology optimization

To ensure manufacturability and mesh independence in density-based topology optimization schemes, it is imperative to use restriction methods. This paper introduces a new class of morphology-based restriction schemes that work as density filters; that is, the physical stiffness of an element is based on a function of the design variables of the neighboring elements. The new filters have the advantage that they eliminate grey scale transitions between solid and void regions. Using different test examples, it is shown that the schemes, in general, provide black and white designs with minimum length-scale constraints on either or both minimum hole sizes and minimum structural feature sizes. The new schemes are compared with methods and modified methods found in the literature.     

A pseudo-sensitivity based discrete-variable approach to structural topology optimization with multiple materials

An algorithm has been developed which uses material as a discrete variable in multi-material topology optimization and thus provides an alternative to traditional methods using material interpolation and level-set approaches. The algorithm computes ‘pseudo-sensitivities’ of the objective and constraint functions to discrete material changes. These are used to rank elements, based on which a fraction of elements are selected for material ID modification during the optimization iteration. The algorithm is of general applicability and avoids frequent matrix factorizations so that it is applicable to large structural problems. In addition to the conventionally used evolutionary and morphogenesis approaches for iteration, a new iteration scheme of ‘resubstitution’ which combines the two approaches is presented. The application and functioning of the algorithm is demonstrated through case studies and comparisons with a few benchmark problems, showing its capability in providing mass-optimal topologies under stiffness constraints for various structural problems where multiple materials are considered.     

Layout optimization for multi-bi-modulus materials system under multiple load cases

An optimization model is presented for obtaining optimal layout of multiple bi-modulus materials systems under multiple load cases (MLC). In the optimization model, the objective function is the linearly weighted structural compliance under MLC. The bi-modulus materials in a finite element are replaced by isotropic materials according to the stress state of that element. The equivalent mechanical properties of an element are expressed as the power–law function of the volume fractions (design variables) and moduli of the solid phases. Numerical experiments are presented to verify the validity and efficiency of the present algorithm. The effects of factors including the bi-modulus behavior of materials, the load directions and the weighting schemes of MLC are also investigated numerically.     

Fuzzy tolerance multilevel approach for structural topology optimization

This paper presents a novel methodology, fuzzy tolerance multilevel programming approach, for applying fuzzy set theory and sequence multilevel method to multi-objective topology optimization problems of continuum structures undergoing multiple loading cases. Ridge-type nonlinear membership functions in fuzzy set theory are applied to embody fuzzy and uncertain characteristics essentially involved by the objective and constraint functions. Sequence multilevel method is used to characterize the different priorities of loading cases at different levels making contribution to the final optimum solution, which is practically beneficial to reduce the subjective influence transferred by using weighted approaches. The solid isotropic material with penalization (SIMP) is adopted as the density-stiffness interpolation scheme to relax the original optimization problem and indicate the dependence of material properties with element pseudo-densities. Sequential linear programming (SLP) is used as the optimizer to solve the multi-objective optimization problem formulated using fuzzy tolerance multilevel programming scheme. Numerical instabilities, such as checkerboards and mesh dependencies are summarized and a duplicate sensitivity filtering method, in favor of contributing to the mesh-dependent optimum designs, is subsequently proposed to regularize the singularity of the optimization problem. The validation of the methodologies presented in this work has been demonstrated by detailed examples of numerical applications.     

A note on truss topology optimization under self-weight load: mixed-integer second-order cone programming approach

This study compares the performance of popular sampling methods for computer experiments using various performance measures to compare them. It is well known that the sample points, in the design space located by a sampling method, determine the quality of the meta-model generated based on expensive computer experiment (or simulation) results obtained at sample (or training) points. Thus, it is very important to locate the sample points using a sampling method suitable for the system of interest to be approximated. However, there is still no clear guideline for selecting an appropriate sampling method for computer experiments. As such, a sampling method, the optimal Latin hypercube design (OLHD), has been popularly used, and quasi-random sequences and the centroidal Voronoi tessellation (CVT) have begun to be noticed recently. Some literature on the CVT asserted that the performance of the CVT was better than that of the LHD, but this assertion seems unfair because those studies only employed space-filling performance measures in favor of the CVT. In this research, we performed the comparison study among the popular sampling methods for computer experiments (CVT, OLHD, and three quasi-random sequences) with employing both space-filling properties and a projective property as performance measures to fairly compare them. We also compared the root mean square error (RMSE) values of Kriging meta-models generated using the five sampling methods to evaluate their prediction performance. From the comparison results, we provided a guideline for selecting appropriate sampling methods for some systems of interest to be approximated.     

Generalized shape optimization using stress constraints under multiple load cases

Optimal shape design using numerical techniques is an increasingly useful engineering tool. Generalized or layout optimal design where the topology of the object is not fixed is one of the emerging applications. These problems are numerically difficult to solve due to the large number of design variables and equality/inequality constraints. Solutions have focused primarily on compliance based minimization under a fixed volume. A more usual engineering approach would be one of minimizing the volume under a stress or deflection constraint. This, however, can lead to problems as stress is a local quantity and volume minimization of multiple load cases under stress constraints may not result in the stiffest design for the remaining material. The approach adopted here is based on a differential rate equation governed by a local operator that defines the state of each element at each time step. This algorithm forms the optimality criteria for the problem. To satisfy the global stress constraints, a feedback derivative is used, analogous to a Lagrange multiplier. The original method for a single load case developed by these authors is extended to deal with multiple load cases. Additionally, a discussion of the global behaviour is included.     

Optimality criteria method for topology optimization under multiple constraints

The existing framework of optimality criteria method is limited to the optimization of a simple energy functional with a single constraint on material resource. The present work extends the optimality criteria method to the case of multiple constraints. The difficulty in updating the Lagrangian multipliers is treated by gradient-split Taylor series expansion. Applications of the method are illustrated by computing the optimal structures under multiple displacement constraints, and by designing the material cells under given macroscopic elastic tensors that correspond to both positive and negative Poisson's ratios.     

Stochastic structural topology optimization: discretization and penalty function approach

We consider structural topology optimization problems, including unilateral constraints arising from, for example, non-penetration conditions in contact mechanics or non-compression conditions for elastic ropes. To construct more realistic models and to circumvent possible failures or inefficient behaviour of optimal structures, we allow parameters (for example, loads) defining the problem to be stochastic. The resulting non-smooth stochastic optimization problem is an instance of stochastic mathematical programs with equilibrium constraints (MPEC), or stochastic bilevel programs. We propose a solution scheme based first on the approximation of the given topology optimization problem by a sequence of simpler sizing optimization problems, and second on approximating the probability measure in the latter problems. For stress-constrained weight-minimization problems, an alternative to -perturbation based on a new penalty function is proposed.     

Reliability-based structural optimization of frame structures for multiple failure criteria using topology optimization techniques

Topology optimization methods using discrete elements such as frame elements can provide useful insights into the underlying mechanics principles of products; however, the majority of such optimizations are performed under deterministic conditions. To avoid performance reductions due to later-stage environmental changes, variations of several design parameters are considered during the topology optimization. This paper concerns a reliability-based topology optimization method for frame structures that considers uncertainties in applied loads and nonstructural mass at the early conceptual design stage. The effects that multiple criteria, namely, stiffness and eigenfrequency, have upon system reliability are evaluated by regarding them as a series system, where mode reliabilities can be evaluated using first-order reliability methods. Through numerical calculations, reliability-based topology designs of typical two- or three-dimensional frames are obtained. The importance of considering uncertainties is then demonstrated by comparing the results obtained by the proposed method with deterministic optimal designs.     

A general formulation of structural topology optimization for maximizing structural stiffness

This paper presents a general formulation of structural topology optimization for maximizing structure stiffness with mixed boundary conditions, i.e. with both external forces and prescribed non-zero displacement. In such formulation, the objective function is equal to work done by the given external forces minus work done by the reaction forces on prescribed non-zero displacement. When only one type of boundary condition is specified, it degenerates to the formulation of minimum structural compliance design (with external force) and maximum structural strain energy design (with prescribed non-zero displacement). However, regardless of boundary condition types, the sensitivity of such objective function with respect to artificial element density is always proportional to the negative of average strain energy density. We show that this formulation provides optimum design for both discrete and continuum structures.     

Preference-based topology optimization for vehicle concept design with concurrent static and crash load cases

In the simulation-based design process of automotive structures, an increasing amount of multi-disciplinary requirements have to be considered. Methods of topology optimization can be used to devise structural concepts early in the design process to obtain the best possible structural layout as starting point for further development steps. Especially relevant for the vehicle design process is the concurrent consideration of static load requirements representing normal operating conditions and energy absorption requirements targeting passive safety in crash events. When the disciplines are considered separately, the heuristic Hybrid Cellular Automaton topology optimization is a suitable method. However, in practical applications, both disciplines are usually addressed sequentially. This complicates the overall process and may reduce the quality of the final optimization result, since optimization objectives may be conflicting. We propose a preference-based Scaled Energy Weighting approach to address the topology optimization of both disciplines concurrently. The main idea is to decouple the user preference from the scaling of the different magnitudes of energies. This enables a multi-objective optimization and ultimately the selection of the desired trade-off solution. We first validate the capability of the method to provide structures optimized for stiffness and energy absorption objectives on beam examples. Finally, the method is applied to optimize a concept structure of an industrial vehicle body, demonstrating its practical feasibility.     

A lower-bound formulation for the geometry and topology optimization of truss structures under multiple loading

In this contribution, we propose an effective formulation to address the stress-based minimum volume problem of truss structures. Starting from the lower-bound formulation in topology optimization, the problem is further expanded to geometry optimization and multiple loading scenarios, and systematically reformulated to alleviate numerical difficulties related to the melting node effect and stress singularities. The subsequent simultaneous analysis and design (SAND) formulation is well suited for a direct treatment by introducing a barrier function. Using exact second derivatives, this difficult class of problem is solved by sequential quadratic programming with trust regions. These building blocks result into an integrated design process. Two examples–including a large-scale application–illustrate the robustness of the proposed formulation.     

A unified aggregation and relaxation approach for stress-constrained topology optimization

In this paper, we propose a unified aggregation and relaxation approach for topology optimization with stress constraints. Following this approach, we first reformulate the original optimization problem with a design-dependent set of constraints into an equivalent optimization problem with a fixed design-independent set of constraints. The next step is to perform constraint aggregation over the reformulated local constraints using a lower bound aggregation function. We demonstrate that this approach concurrently aggregates the constraints and relaxes the feasible domain, thereby making singular optima accessible. The main advantage is that no separate constraint relaxation techniques are necessary, which reduces the parameter dependence of the problem. Furthermore, there is a clear relationship between the original feasible domain and the perturbed feasible domain via this aggregation parameter.     

A comparative study of dynamic analysis methods for structural topology optimization under harmonic force excitations

This work is focused on the topology optimization related to harmonic responses for large-scale problems. A comparative study is made among mode displacement method (MDM), mode acceleration method (MAM) and full method (FM) to highlight their effectiveness. It is found that the MDM results in the unsatisfactory convergence due to the low accuracy of harmonic responses, while MAM and FM have a good accuracy and evidently favor the optimization convergence. Especially, the FM is of superiority in both accuracy and efficiency under the excitation at one specific frequency; MAM is preferable due to its balance between the computing efficiency and accuracy when multiple excitation frequencies are taken into account.     

A systematic topology optimization approach for optimal stiffener design

A systematic topology optimization based approach is proposed to design the optimal stiffener of three-dimensional shell/plate structures for static and eigenvalue problems. Optimal stiffener design involves the determination of the best location and orientation. In this paper, the stiffener location problem is solved by a microstructure-based design domain method and the orientation problem is modelled as an optimization orientation problem of equivalent orthotropic materials, which is solved by a newly developed energy-based method. Examples are presented to demonstrate the application of the proposed approach.     

A new level-set based approach to shape and topology optimization under geometric uncertainty

Geometric uncertainty refers to the deviation of the geometric boundary from its ideal position, which may have a non-trivial impact on design performance. Since geometric uncertainty is embedded in the boundary which is dynamic and changes continuously in the optimization process, topology optimization under geometric uncertainty (TOGU) poses extreme difficulty to the already challenging topology optimization problems. This paper aims to solve this cutting-edge problem by integrating the latest developments in level set methods, design under uncertainty, and a newly developed mathematical framework for solving variational problems and partial differential equations that define mappings between different manifolds. There are several contributions of this work. First, geometric uncertainty is quantitatively modeled by combing level set equation with a random normal boundary velocity field characterized with a reduced set of random variables using the Karhunen–Loeve expansion. Multivariate Gauss quadrature is employed to propagate the geometric uncertainty, which also facilitates shape sensitivity analysis by transforming a TOGU problem into a weighted summation of deterministic topology optimization problems. Second, a PDE-based approach is employed to overcome the deficiency of conventional level set model which cannot explicitly maintain the point correspondences between the current and the perturbed boundaries. With the explicit point correspondences, shape sensitivity defined on different perturbed designs can be mapped back to the current design. The proposed method is demonstrated with a bench mark structural design. Robust designs achieved with the proposed TOGU method are compared with their deterministic counterparts.