Entropy-regularized Wasserstein distributionally robust shape and topology optimization

This paper presents a mixed integer programming (MIP) formulation for robust topology optimization of trusses subjected to the stress constraints under the uncertain load. A design‐dependent uncertainty model of the external load is proposed for dealing with the variation of truss topology in the course of optimization. For a truss with the discrete member cross‐sectional areas, it is shown that the robust topology optimization problem can be reduced to an MIP problem, which is solved globally. Numerical examples illustrate that the robust optimal topology of a truss depends on the magnitude of uncertainty. Copyright © 2010 John Wiley & Sons, Ltd.

Robust topology and discrete fiber orientation optimization under principal material uncertainty

The robust topology optimization (RTO) design of multi-material structures holds significant practical implications and theoretical value. The main contribution of this paper is to study the RTO problem of multi-material structures considering load uncertainty, and to propose an innovative non-gradient multi-material RTO approach to multi-material structures considering load uncertainty. To this end, the alternating active-phase algorithm (AAPA) is first applied to decouple this RTO problem, and combined with these approaches (probabilistic approach, density-based approach, and linear combination approach) to construct the RTO model of multi-material structures considering load uncertainty and taking the linear combination of the mean and standard deviation of the structural compliance as the objective function. Subsequently, the superposition principle of linear theory, Monte Carlo simulation (MCS), and orthogonal diagonalization of a symmetric matrix are applied to derive the computational equation for the objective function suitable for an improved proportional topology optimization (IPTO) approach. Based on this, the IPTO approach is employed to solve this RTO model, thereby forming an innovative multi-material non-gradient RTO approach. Finally, two numerical examples are applied to illustrate the efficacy of this new approach and the influence of the filtering radius on the structural RTO design. The results showcase that this new approach can tackle the robust optimization design issues of multi-material structures and yield an optimal structure with enhanced robustness. Moreover, the proposed approach's filtering radius directly influences the shape of the optimized structure.

Robust topology optimization of multi-material structures under load uncertainty using the alternating active-phase method

Efficient robust topology optimization of piezoelectric energy harvesters with smooth boundaries and reduced computational cost

Topology optimization under uncertainty poses extreme difficulty to the already challenging topology optimization problem. This paper presents a new computational method for calculating topological sensitivities of statistical moments of high-dimensional complex systems subject to random inputs. The proposed method, capable of evaluating stochastic sensitivities for large-scale, robust topology optimization (RTO) problems, integrates a polynomial dimensional decomposition (PDD) of multivariate stochastic response functions and deterministic topology derivatives. In addition, the statistical moments and their topology sensitivities are both determined concurrently from a single stochastic analysis. When applied in collaboration with the gradient based optimization algorithm, the proposed method affords the ability of solving industrial-scale RTO design problems. Numerical examples indicate that the new method developed provides computationally efficient solutions.

This paper presents a performance index for topology and shape optimization of plate bending problems with displacement constraints. The performance index is developed based on the scaling design approach. This performance index is used in the Performance-Based Optimization (PBO) method for plates in bending to keep track of the performance history when inefficient material is gradually removed from the design and to identify optimal topologies and shapes from the optimization process. Several examples are provided to illustrate the validity and effectiveness of the proposed performance index for topology and shape optimization of bending plates with single and multiple displacement constraints under various loading conditions. The topology optimization and shape optimization are undertaken for the same plate in bending, and the results are evaluated by using the performance index. The proposed performance index is also employed to compare the efficiency of topologies and shapes produced by different optimization methods. It is demonstrated that the performance index developed is an effective indicator of material efficiency for bending plates. From the manufacturing and efficient point of view, the shape optimization technique is recommended for the optimization of plates in bending.

Complex uncertainty-oriented robust topology optimization for multiple mechanical metamaterials based on double-layer mesh

Structural topology optimization.- On basic properties of Michell's structures.- Validation of numerical method by analytical benchmarks and verification of exact solutions by numerical methods.- Introduction to shape and topology optimization.- Homogenization method for shape and topology optimization.- Level set method for shape and topology optimization.- Compliance minimization of two-material elastic structures.- Some notes on topology optimization of vibrating continuum structures.- Topology optimization of vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps.- On optimum shape design and periodicity of band-gape beam structures.- Topological design for minimum dynamic compliance of continuum structures subjected to forced vibration.- Topological design for minimum sound emission from bi-material structures subjected to forced vibration.- Discrete material optimization of vibrating laminated composite plates for minimum sound emission.- Optimization of diffusive transport problems.- Fluid topology optimization: stokes and Navier-stokes models.- Topology optimization of coupled multi-physics problems.- Topology optimization based on the extended finite element method.- Topology optimization of meso- and nano-scale problems.- Topology optimization under uncertainty.- A brief review of numerical methods of structural topology optimization.- The free material design in planar elasticity.

This paper presents a worst case approach for robust geometry and topology optimization of plane frames with global stability constraint. Uncertainty is assumed to exist in the nodal locations and cross‐sectional areas, and the worst values of the objective and stability constraint functions are relaxed to the quantile structural responses represented by the order statistics with given robustness and confidence levels. In order to alleviate the difficulty caused by melting nodes to some extent, the force density method is applied to an auxiliary truss model for geometry optimization of the frame, and the closely spaced nodes are merged. A method is presented for generating correlated imperfections for the nodal locations along each member, and a penalization approach is proposed for geometrical stiffness matrix to exclude superficial local buckling. It is demonstrated in the numerical examples that the result of robust optimization obtained by the proposed method is less sensitive to the uncertainty, and the stability constraint is also satisfied under uncertainty with the specified robustness and confidence levels.

In order to achieve a low vibration design of engine block,verify the correctness of finite element model based on the test method,ensure the undertake of different levels of optimization model using the method of topology and shape optimization,modal frequency and vibration intensity of engine block are taken as optimization objectives,vibration analysis of engine block is made in the simulation,and the body vibration condition distribution are identified in the optimization design.The four cylinder engine block is set as the design objective,the optimal solution design variables are got through the topology optimization and shape optimization method,and then the final optimization scheme with whole layout and detailed design are obtained.The results of optimized structure show that frequencies of each mode are improved,and the vibration intensity value of engine block decreases by 34.8%,then vibration intensity of improved engine block is minimized in a safe area subject to rigidity and strength constraints.Efficiency and precision of the combination method of topology and shape optimization has an obvious enhance,and the feasibility and efficiency of the proposed methods are demonstrated by the experiment.

Integrated topology and shape optimization software for compliant MEMS mechanism design

This paper deals with simultaneous topology and shape optimization of elastic contact problems. The structural optimization problem for an elastic contact problem is formulated. Shape as well as topological derivatives formulae of the cost functional are provided using material derivative and asymptotic expansion methods, respectively. These derivatives are employed to formulate necessary optimality condition for simultaneous shape and topology optimization and to calculate a descent direction in numerical algorithm. Level set based numerical algorithm for the solution of this optimization problem is proposed. Numerical examples are provided and discussed.

Robust structural topology optimization considering boundary uncertainties

Topology optimization of structures with interval random parameters