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One of the biggest challenges in optimizing the shape of complex solids is the requirement to maintain a reasonable mesh quality not only at the boundary but also for the bulk discretization of the interior. Thus, additional regularization and, in many cases, re-meshing of the structure during the iterative process is unavoidable with a Lagrangian description. By tracking the shape update using an Eulerian representation, embedded boundary methods are a promising technique for eliminating mesh distortion problems. This work consistently combines the unique features of implicit Vertex-Morphing and embedded boundary methods, facilitating the node-based shape optimization of solids with industrial complexity. One of the crucial elements for solving the primal problem on a fixed background grid is an efficient and robust quadrature scheme. To this end, we incorporate the open-source C++ framework QuESo (https://github.com/manuelmessmer/QuESo) developed for the numerical integration of arbitrarily complex embedded solids defined by oriented boundary meshes, e.g., in STereoLithography (STL) format. Meanwhile, applying the Helmholtz/Sobolev-based (implicit) filter to the vertices of the embedded boundary mesh not only exploits the extensive design space of node-based optimization but also ensures robust control over the feature size. To realize the above methodology, this work introduces a novel sensitivity analysis that yields mesh-independent shape gradients with respect to the immersed boundary. Through a specific sensitivity weighting, we recover a continuous gradient field from discrete values calculated at the nodes of the background grid. In addition, the presented workflow ensures robust enforcement of challenging geometrical constraints, including minimum wall thicknesses and design space limitations. We critically assess our approach with benchmarks and structures of industrial relevance. In all examples, trivariate B-Spline bases span the background grid, providing highly accurate finite element solutions and shape sensitivities at every iteration. Moreover, the elimination of mesh distortion problems enables a successful termination at the local optimum, even for large shape modifications. • Extension of Vertex-Morphing to embedded boundary methods. • Rigorous elimination of mesh distortion problems. • Discretization-independent sensitivity analysis. • Robust enforcement of design space constraints. • Application to solid structures with industrial complexity. [ABSTRACT FROM AUTHOR]

This article investigates the convergence properties of the augmented finite element method (AFEM). The AFEM is here used to model weak discontinuities independently of the underlying mesh. One noticeable advantage of the AFEM over other partition of unity methods is that it does not introduce additional global unknowns. Numerical 2D experiments illustrate the performance of the method and draw comparisons with the finite element method (FEM) and the nonconforming FEM. It is shown that the AFEM converges with an error of 𝒪(h0.5) in the energy norm. The nonconforming FEM shares the same property while the FEM converges at 𝒪(h). Yet, the AFEM is on par with the FEM for certain homogenization problems. [ABSTRACT FROM AUTHOR]

Embedded and immersed methods have become essential tools in computational mechanics, as they allow discretizing arbitrarily complex geometries without the need for boundary-fitted meshes. One of their main challenges is the accurate numerical integration of cut elements. Among the various integration schemes developed for this purpose, moment fitting has proven to be a powerful technique that provides highly efficient and accurate integration rules. This publication presents a framework for the robust and efficient numerical integration of embedded solids described by oriented boundary meshes using moment fitting. The developments include an intersection algorithm that aims to drastically accelerate the computation of the necessary moments while achieving high accuracy. A closed surface parameterization of each cut domain is computed to facilitate the direct application of the divergence theorem. The algorithm is subject to a single quality criterion that guarantees the accurate evaluation of boundary integrals. At the same time, it allows to disregard classical mesh criteria, such as high aspect ratios, strongly varying angles, etc., resulting in extremely fast runtimes. In addition, an existing robust flood fill-based element classification scheme is further developed to initiate filling from arbitrary seed elements and to enable parallel execution, increasing its flexibility and efficiency. The successful application of all proposed algorithms to 4948 valid and flawed STLs from the Thingi10K database (Zhou and Jacobson, 2016) demonstrates their extraordinary robustness. In all cases, the wall-clock time scales at most linearly with the number of elements in the background mesh. We show that higher-order quadrature rules on the boundary elements enable efficient computation of the moments via the divergence theorem with near-machine precision. Finally, the presented methodologies are used to perform direct FE analyses on clean and flawed B-Rep models. All proposed algorithms are publicly available in the open-source C++ framework QuESo – Quadrature for Embedded Solids (https://github.com/manuelmessmer/QuESo), where the moment fitting equations are assembled and solved. • Fast assembly of the moment fitting equations for arbitrarily complex faceted B-Reps. • Efficient intersection algorithm to facilitate the application of the divergence theorem. • Generalized and parallel-executable element classification scheme based on flood filling. • Successful application of the framework to 4948 valid and flawed STLs. [ABSTRACT FROM AUTHOR]

This paper investigates the accuracy and the convergence properties of the augmented finite element method (AFEM). The AFEM is here used to model strong discontinuities independently of the underlying mesh. One noticeable advantage of the AFEM over other partition of unity methods is that it does not introduce additional global unknowns to represent cracks. Numerical 2D experiments illustrate the performance of the method and draw comparisons with the element deletion method (EDM), the phantom node method (PNM), the finite element method (FEM) and the embedded finite element method (EFEM). The h-convergence in the energy norm of the AFEM is studied for the first time and it is shown to outperform the aforementioned numerical methods when cracks are loaded in Mode I. • The convergence rate of the AFEM is numerically evaluated. • The singularity of AFEM stiffness matrices is dealt with. • The accuracy of the AFEM is thoroughly benchmarked against popular numerical methods. • The AFEM outperforms the other evaluated methods under Mode I loading of a crack. [ABSTRACT FROM AUTHOR]