For aeronautical applications of topology optimization, it is of importance to develop topology optimization techniques that can handle stress constraints in an efficient and accurate manner. The development of such topology optimization techniques is a challenging task due to the local nature of the stress constraints, their highly non-linear behaviour with respect to the design variables and the so-called singularity phenomenon. An accurate sensitivity analysis is essential for these type of problems with multiple constraints. In this paper, we propose a methodology of dealing with stress constraints in a level set based framework. In this framework, the level set function nodal values are related to element densities by an exact Heaviside projection. Stress relaxation and constraint aggregation techniques are used to deal with the singularity phenomenon and the local nature of the stress, respectively. A constrained optimization problem is then solved, in which the design variables (the level set nodal values) are updated in the projected steepest-descent direction, which is determined using a consistent sensitivity analysis. We demonstrate the effe4ctiveness of this technique on two numerical examples. The results show that the level set method with a consistent sensitivity analysis allows for the treatment of multiple constraints by using constrained optimization techniques.@en

This paper presents a level-set-based topology optimization method based on numerically consistent sensitivity analysis. The proposed method uses a direct steepest-descent update of the design variables in a level-set method; the level-set nodal values. An exact Heaviside formulation is used to relate the level-set function to element densities. The level-set function is not required to be a signed-distance function, and reinitialization is not necessary. Using this approach, level-set-based topology optimization problems can be solved consistently and multiple constraints treated simultaneously. The proposed method leads to more insight in the nature of level-set-based topology optimization problems. The level-set-based design parametrization can describe gray areas and numerical hinges. Consistency causes results to contain these numerical artifacts. We demonstrate that alternative parameterizations, level-set-based or density-based regularization can be used to avoid artifacts in the final results. The effectiveness of the proposed method is demonstrated using several benchmark problems. The capability to treat multiple constraints shows the potential of the method. Furthermore, due to the consistency, the optimizer can run into local minima; a fundamental difficulty of level-set-based topology optimization. More advanced optimization strategies and more efficient optimizers may increase the performance in the future. Key Words: topology optimization; level-set method; consistency; sensitivity analysis, exact Heaviside; regularization@en

In this paper, we address a newly discovered local instability observed in geometrically nonlinear analysis using the Internal Element Connectivity Parameterization (I-ECP) approach. I-ECP offers advantages in topology optimization of nonlinear elastic structures compared to conventional material scaling approaches (e.g. SIMP), including more robustness against excessive deformations. However, under certain conditions, we observe local instabilities in low-density areas using I-ECP as well. These instabilities can result in analysis failure and premature termination of the topology optimization process. We report on this finding and present a small numerical example exhibiting the undesired behavior. Furthermore, we provide insight into its origin on a fundamental level based on an I-ECP truss as well as using the general I-ECP governing equations. In conclusion, when sufficient support of surrounding elements is lacking, highly compressed I-ECP patches can trigger a buckling mode on system level. Our analysis reveals that the element-link arrangement employed in I-ECP results in a magnification of negative eigenvalues, which promotes the instability. In addition to an investigation of this phenomena, we provide various directions for stabilization. Keywords: Topology optimization, Internal Element Connectivity Parameterization, geometrical nonlinearity, material nonlinearity, instability.@en

This contribution presents a novel and versatile approach to geometrically nonlinear topology optimization by combining the level-set method with the element connectivity parameterization method or ECP. The combined advantages of both methods open up the possibility to treat a wide range of optimization problems involving complex physical and/or geometrical nonlinearities in a general and elegant manner. The level-set method features shape optimization on a fixed mesh, leading to intrinsically black-and-white designs. This approach allows a clear description of location and orientation of the interface, whereas topological changes can still be handled easily. A popular concept used in conventional level-set methods is to map the level-set function to volume-fraction design variables for every element of a finite element mesh. The resulting element density variables are then used to scale the Young¿s modulus in each element using the Ersatz material approach. In this work we employ a modified material interpolation method, in which the element density variables, based on a per-element integration of a regularized Heaviside operator applied to the level-set function, are used as element connectivity design variables. The resulting crisp boundary topology optimization method exploits the advantages of ECP in the field of complex nonlinearities and eliminates the need for penalization by the implicit level-set description of the design.@en

In the past decade, SIMP-based topology optimization methods have become increasingly popular. They offer the designer the possibility to perform design optimization in an early stage of the design process for a wide range of applications. The specific design parameterization used in SIMP is the result of many attempts to eliminate numerical artifacts in the results of topology optimization problems. However, any choice of design parameterization also influences the shape of the response functions. For a gradient-based optimization method it is important to obtain a smooth and, if possible, convex response. This research aims to point out relevant consequences of using the SIMP design parameterization in combination with density filters on the response functions. In this contribution we show the effects of the power of the penalization and the size of the density filter on the response functions. The introduction of penalization has two effects. Firstly, increasing the introduction of a density filter is not only useful to avoid checkerboard patterns, control minimum member size and achieve mesh-independence, but it also reduces the immobilizing effect that penalization has on the mobility of the boundary of the structure. This effect and its dependence on the power of the penalization and the size of the density filter are illustrated using several elementary numerical examples involving compliance minimization. The results are expected to be relevant for any type of structural topology optimization. Keywords: Topology optimization, design parameterization, penalization, density filter, local minima@en

Level-set based topology optimization approaches are gaining popularity due to their clear design description and potential for more natural treatment of multi-physics problems involving interfaces between physical domains.@en

This contribution presents a novel and versatile approach to geometrically nonlinear topology optimization by combining the level-set method with the element-connectivity-parameterization method (ECP). The combined advantages of both methods open up the possibility to treat a wide range of optimization problems involving complex physical and/or geometrical nonlinearities in a general and elegant way. The level-set method features shape optimization on a fixed mesh, leading to intrinsically black-and-white designs. This approach allows a clear description of location and orientation of the interface, whereas topological changes can still be handled easily. A popular concept used in conventional level-set methods is to map the level-set function to volume-fraction design variables for every element of a finite element mesh using the Ersatz-material approach. In this work we employ a modified Ersatz-material formulation, in which the element-connectivity design variables are based on a per-element integration of a regularized Heaviside operator applied to the level-set function. The resulting crisp-boundary topology optimization method exploits the abilities of ECP in the field of complex nonlinearities and eliminates the need for penalization by the implicit level-set description of the design. Keywords: Topology optimization, level-set method, element-connectivity-parameterization method (ECP), geometrical nonlinearities.@en