To Örebro University

In this paper, we investigate the finite element discretization of the isotropic Helmholtz-type PDE filter for topology optimization and propose a robust alternative based on mass lumping. While the new strategy preserves the filter’s efficiency, it enforces the discrete maximum principle, ensuring that the filtered design variables remain strictly within the admissible interval. We establish theoretical support for the approach, and comprehensive numerical experiments validate its reliability and effectiveness. Numerical results show that the proposed approach provides virtually identical results to those of the standard PDE filter for radii larger than the characteristic mesh size. It also delivers improved accuracy and stability when the radius is comparable to or smaller than the characteristic mesh size. In topology optimization, this may lead to distinct designs and provide enhanced numerical stability within this range.

Incorporating stress constraints in topology optimization is a challenging task due to the large number of constraints in the formulation. One effective strategy to address this challenge is the augmented Lagrangian (AL) method, which transforms the original stress-constrained problem into a sequence of subproblems with only bound constraints. The effectiveness of the AL method heavily depends on the optimization method used to solve these subproblems. This work performs a comparative study of six optimization solvers: The method of moving asymptotes (MMA), the steepest descent method with move limits (SDM), the spectral projected gradient (SPG), and the limited-memory BFGS with bound constraints (L-BFGS-B), along with two proposed adaptations, steepest descent method with move limits—Barzilai–Borwein (SDMBB) and spectral projected gradient with move limits (SPGM). These methods are evaluated in the context of the volume minimization problem with local stress constraints. The solutions are compared in terms of performance, defined as the final volume fraction, and efficiency, measured by the number of state and adjoint analyses required. A mesh dependence study is conducted to assess the robustness of each method across different mesh sizes, including high-resolution cases with approximately 1.8 million elements. SDMBB exhibits the highest efficiency, while SPGM achieves the best performance, followed by SDM. The MMA, SPG, and L-BFGS-B show limitations in high-resolution problems or fail to meet specific stopping criteria. The results demonstrate that the choice of the optimization solver significantly affects the efficiency of the AL method, as well as the performance and mesh dependence of the solutions. Furthermore, this study identifies the most promising methods for solving large-scale stress-constrained problems.

This work proposes a stress-constrained topology optimization formulation to design minimum volume structures that are both fail-safe and manufacturing error tolerant. The goal is to obtain an optimized topology that satisfies the stress failure criterion before and after the occurrence of any damage, also taking into account slight imperfections that may occur during the structure manufacturing process. In order to achieve this goal, the conventional volume minimization formulation with local stress constraints is generalized to simultaneously accommodate possible damage scenarios, considering a simplified damage model of removal of failure regions with predefined shape, size and position from the design domain, and possible manufacturing scenarios, which are included via the three-field density projection approach based on intermediate, eroded and dilated designs. Two numerical examples are addressed: L-shaped and Cantilever problems; both are solved for different arrangements of damage scenarios. Stress constraints are handled via the augmented Lagrangian method, without using aggregation techniques, and problems with up to 82.5 million stress constraints are solved. Numerical investigations demonstrate that: (1) the proposed formulation is able to obtain results which are at the same time fail-safe and tolerant to uniform manufacturing error, as for any manufacturing situation the maximum stress does not exceed the allowable design stress after the occurrence of any predefined damage; (2) when manufacturing error tolerance is not considered, extremely sensitive results are obtained, being truly fail-safe for unique manufacturing situations.

This paper addresses the concept of predetermined breaking points in topology optimization. The aim is to propose and investigate a novel formulation to design optimized topologies in which one can control where failure will occur first in case of overload; in addition, the optimized topology must withstand the design load after the damaged part is removed. In order to achieve this goal, a stress-constrained formulation based on two realizations of material distributions is proposed: one realization represents the nominal design, without damage, and the other represents the damaged design. In the nominal design, the predetermined damage region is defined, which is the region where failure is programmed to occur first in case of overload. The design constraints are defined in a way that ensures that a structural member is formed within the predetermined damage region and that the maximum von Mises equivalent stress of this member is slightly larger than the maximum von Mises stress in the rest of the structure. After failure has occurred, stress constraints are employed to ensure that the resulting design without the damaged part still resists the applied load. Two design problems with several variants are addressed: the L-shaped and the MBB beam problems. Numerical investigations demonstrate that: (1) the conventional design is extremely sensitive to localized damage of structural members and, moreover, its almost fully stressed configuration does not allow to predict where failure will occur first in case of overload; (2) the proposed formulation for predetermined breaking points is able to provide optimized structures where one knows in advance the region where failure is expected to occur first; in addition, the structure remains safe after the damaged part is removed.

Stress-constrained topology optimization requires techniques for handling thousands to millions of stress constraints. This work presents a comprehensive numerical study comparing local and global stress constraint strategies in topology optimization. Four local and four global solution strategies are presented and investigated. The local strategies are based on either the augmented Lagrangian or the pure exterior penalty method, whereas the global strategies are based on the P-mean aggregation function. Extensive parametric studies are carried out on the L-shaped design problem to identify the most promising parameters for each solution strategy. It is found that (1) the local strategies are less sensitive to the continuation procedure employed in standard density-based topology optimization, allowing achievement of better quality results using less iterations when compared with the global strategies; (2) the global strategies become competitive when P values larger than 100 are employed, but for this to be possible a very slow continuation procedure should be used; (3) the local strategies based on the augmented Lagrangian method provide the best compromise between computational cost and performance, being able to achieve optimized topologies at the level of a pure P-continuation global strategy with P=300, but using less iterations.

It is nowadays widely acknowledged that optimal structural design should be robust with respect to the uncertainties in loads and material parameters. However, there are several alternatives to consider such uncertainties in structural optimization problems. This paper presents a comprehensive comparison between the results of three different approaches to topology optimization under uncertain loading, considering stress constraints: (1) the robust formulation, which requires only the mean and standard deviation of stresses at each element; (2) the reliability-based formulation, which imposes a reliability constraint on computed stresses; (3) the non-probabilistic formulation, which considers a worst-case scenario for the stresses caused by uncertain loads. The information required by each method, regarding the uncertain loads, and the uncertainty propagation approach used in each case is quite different. The robust formulation requires only mean and standard deviation of uncertain loads; stresses are computed via a first-order perturbation approach. The reliability-based formulation requires full probability distributions of random loads, reliability constraints are computed via a first-order performance measure approach. The non-probabilistic formulation is applicable for bounded uncertain loads; only lower and upper bounds are used, and worst-case stresses are computed via a nested optimization with anti-optimization. The three approaches are quite different in the handling of uncertainties; however, the basic topology optimization framework is the same: the traditional density approach is employed for material parameterization, while the augmented Lagrangian method is employed to solve the resulting problem, in order to handle the large number of stress constraints. Results are computed for two reference problems: similarities and differences between optimized topologies obtained with the three formulations are exploited and discussed.

In topology optimization, the treatment of stress constraints for very large scale problems (more than 100 million elements and more than 600 million stress constraints) has so far not been tractable due to the failure of robust agglomeration methods, i.e. their inability to accurately handle the locality of the stress constraints. This paper presents a three‐dimensional design methodology that alleviates this shortcoming using both deterministic and robust problem formulations. The robust formulation, based on the three‐field density projection approach, is extended and proved necessary to handle manufacturing uncertainty in three‐dimensional stress‐constrained problems. Several numerical examples are solved and further post‐processed with body‐fitted meshes using commercial software. The numerical investigations demonstrate that: (1) the employed solution approach based on the augmented Lagrangian method is able to handle very large problems, with hundreds of millions of stress constraints; (2) three‐dimensional stress‐based results are extremely sensitive to slight manufacturing variations; (3) if appropriate interpolation parameters are adopted, voxel‐based (fixed grid) models can be used to compute von Mises stresses with excellent accuracy; and (4) in order to ensure manufacturing tolerance in threedimensional stress‐constrained topology optimization, a combination of double filtering and more than three density field realizations may be required.

This paper proposes and investigates two formulations to topology optimization of compliant mechanisms considering stress constraints, manufacturing uncertainty and geometric nonlinearity. The first formulation extends the maximum output displacement robust approach with stress constraints to incorporate the effects of geometric nonlinear behavior during the optimization process. The second formulation relies on the concept of path-generating mechanisms, where not only the final configuration is important, but also the load–displacement equilibrium path. A novel path-generating formulation is thus proposed, not only to achieve the prescribed equilibrium path, but also to take stress constraints and manufacturing uncertainty into account during the optimization process. Although both formulations have different goals, the same main techniques are employed: density approach to topology optimization, augmented Lagrangian method to handle the large number of stress constraints, three-field robust approach to handle the manufacturing uncertainty, and the energy interpolation scheme to handle convergence issues due to large deformation in void regions. Several numerical examples are addressed to demonstrate applicability of the proposed approaches. The optimized results are post-processed with body-fitted finite element meshes. Obtained results demonstrate that: (1) the proposed nonlinear analysis based maximum output displacement approach is able to provide solutions with good performance in situations of large displacements, with stress and manufacturing requirements satisfied; (2) the linear analysis based maximum output displacement approach provides optimized topologies that show large stress constraint violations and rapidly varying stress behavior under uniform boundary variation, when these are post-processed with full nonlinear analysis; (3) the proposed path-generating formulation is able to provide solutions that follow the prescribed control points, including stress robustness.

This work addresses the topology optimization approach to design robust compliant mechanisms with respect to uncertainties in the output stiffness, when compared to the traditional deterministic approach. To this end, two formulations are proposed: probabilistic and nonprobabilistic. The probabilistic formulation minimizes a joint objective function of expected output displacement plus a measure of its standard deviations, for given statistical distribution of the output stiffness. The nonprobabilistic formulation is written as minimization of a joint function of the median of output displacements, plus the width of the intervals that contains the extreme values of the output displacements, for a given interval of output stiffness. The Monte Carlo simulation method is used to evaluate expected values and standard deviations of output displacements in the probabilistic formulation and to assess results obtained with the deterministic approach. It is shown that both formulations lead to designs where output displacements are less sensitive to variations of output stiffness when compared to the traditional deterministic approach. Furthermore, as an additional benefit, it is observed that large variations of output stiffness can hinder the appearance of one-node connected hinges, usually found in the deterministic design of compliant mechanisms.

This paper proposes a robust design approach, based on eroded, intermediate and dilated projections, to handle uniform manufacturing uncertainties in stress-constrained topology optimization. In addition, a simple scheme is proposed to increase accuracy of stress evaluation at jagged edges, based on limiting sharpness of the projections to intentionally allow a thin layer of intermediate material between solid and void phases. A reference problem is analyzed through voxel-based finite element models, demonstrating that, in association with a proper choice of stiffness and stress interpolation functions, the proposed scheme can ensure consistent stress magnitude and smooth stress behavior for uniform boundary variation. Optimization problems are solved and post-processing with body-fitted meshes is performed over optimized solutions, demonstrating that: (1) stresses evaluated with voxel-based meshes containing thin soft transition boundaries are consistent with stresses evaluated with body-fitted meshes; and (2) optimized structures are robust with respect to uniform boundary variations.