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A non-intrusive proper generalized decomposition (PGD) strategy, coupled with an overlapping domain decomposition (DD) method, is proposed to efficiently construct surrogate models of parametric linear elliptic problems. A parametric multi-domain formulation is presented, with local subproblems featuring arbitrary Dirichlet interface conditions represented through the traces of the finite element functions used for spatial discretization at the subdomain level, with no need for additional auxiliary basis functions. The linearity of the operator is exploited to devise low-dimensional problems with only few active boundary parameters. An overlapping Schwarz method is used to glue the local surrogate models, solving a linear system for the nodal values of the parametric solution at the interfaces, without introducing Lagrange multipliers to enforce the continuity in the overlapping region. The proposed DD-PGD methodology relies on a fully algebraic formulation allowing for real-time computation based on the efficient interpolation of the local surrogate models in the parametric space, with no additional problems to be solved during the execution of the Schwarz algorithm. Numerical results for parametric diffusion and convection–diffusion problems are presented to showcase the accuracy of the DD-PGD approach, its robustness in different regimes and its superior performance with respect to standard high-fidelity DD methods. [ABSTRACT FROM AUTHOR]

In probabilistic robust optimization problems we generally take as objective function the weighted sum of the expected value and the standard deviation of the performance. Unfortunately, it has been observed that if the weight given to the standard deviation is too high, then the designs obtained may have no physical meaning at all. If this occurs we say that the design obtained is not consistent from the physical point of view. In this work we present a probabilistic robust optimization approach aimed to preserve physical consistency of the problem being addressed. The objective function is written as the p -norm of a vector composed of the expected value and the weighted standard deviation. We then demonstrate that the standard approach is a particular case of this p -norm approach with p = 1. In the proposed approach we take p = 2 and prove that physical consistency is ensured provided the standard deviation is not given more weight than the expected value. A similar proof is not available for the standard approach. This makes statement of the robust optimization problem much easier with the proposed approach. The proposed approach can be adapted into existing computational routines, since the same statistical moments and sensitivities are required. The theoretical results are illustrated in the context of compliance-based topology optimization. In the numerical examples we show that the concept of physical consistency may have a significant impact on the designs obtained. • A discussion on physical consistency of robust optimization is presented. • An approach that is always consistent is proposed. • Theoretical results are illustrated in the context of compliance-based topology optimization. [ABSTRACT FROM AUTHOR]

Abstract We present a stabilized finite element method that generalizes Nitsche's method for enforcing stiff anisotropic cohesive laws with different normal and tangential stiffness. For smaller values of cohesive stiffness, the stabilized method resembles the standard method, wherein the traction on the crack surface is enforced as a Neumann boundary condition. Conversely, for larger values of cohesive stiffness, the stabilized method resembles Nitsche's method, wherein the cohesive law is enforced as a kinematic constraint. We present several numerical examples, in two-dimensions, to compare the performance of the stabilized and standard methods. Our results illustrate that the stabilized method enables accurate recovery of crack-face tractions for stiff isotropic and anisotropic cohesive laws; whereas, the standard method is less accurate due to spurious traction oscillations. Also, the stabilized method could mitigate spurious sensitivity of load–displacement results to displacement increment in mixed-mode fracture simulation, owing to its stability and accuracy. [ABSTRACT FROM AUTHOR]

This work constitutes a further contribution to developing convergent, locking-free virtual element formulations for problems of large-deformation nonlinear elasticity. Earlier work by the authors and others has introduced simple and effective forms of stabilization. This work investigates the approximation properties of the virtual element method for problems with domains having a high aspect ratio, and for alternative forms of stabilization: an existing approach, and two modifications, which account for high element aspect ratio and element distortion. The results in all cases demonstrate the locking-free convergence of the approach, with the newer modifications leading to improvements in accuracy. • A lowest-order displacement-based VEM is formulated for linear- and hyper-elasticity. • A novel approach to the scaling of the stabilization term is proposed and evaluated. • The scaling accounts for the aspect ratio and the degree of element distortion. • The proposed method exhibits superior performance to the standard VEM formulation. • The well-known locking-free behaviour of the lowest-order VEM is preserved. [ABSTRACT FROM AUTHOR]

Bayesian computation plays an important role in modern machine learning and statistics to reason about uncertainty. A key computational challenge in Bayesian inference is to develop efficient techniques to approximate, or draw samples from posterior distributions. Stein variational gradient decent (SVGD) has been shown to be a powerful approximate inference algorithm for this issue. However, the vanilla SVGD requires calculating the gradient of the target density and cannot be applied when the gradient is unavailable or too expensive to evaluate. In this paper we explore one way to address this challenge by the construction of a local surrogate for the target distribution in which the gradient can be obtained in a much more computationally feasible manner. More specifically, we approximate the forward model using a deep neural network (DNN) which is trained on a carefully chosen training set, which also determines the quality of the surrogate. To this end, we propose a general adaptation procedure to refine the local approximation online without destroying the convergence of the resulting SVGD. This significantly reduces the computational cost of SVGD and leads to a suite of algorithms that are straightforward to implement. The new algorithm is illustrated on a set of challenging Bayesian inverse problems, and numerical experiments demonstrate a clear improvement in performance and applicability of standard SVGD. [ABSTRACT FROM AUTHOR]

An immersed enriched finite element method is proposed for the analysis of phononic crystals (PnCs) with finite element (FE) meshes that are completely decoupled from geometry. Particularly, a technique is proposed to prescribe Bloch–Floquet periodic boundary conditions strongly on non-matching edges of the periodic unit cell (PUC). The enriched finite element formulation effectively transforms a periodic non-confirming discretization into an enriched node-to-node periodic discretizations where periodicity is enforced by any standard procedure. The enriched formulation is also used to describe the interior material interface. This completely eliminates the tedious process of generating matching or fitted meshes during the design process, as it allows changing the inclusion's geometry as well as the PnC's lattice type without changing the FE mesh. The proposed approach, which is used to analyze phononic crystals in 1-D, 2-D, and 3-D using structured meshes, exhibits the same performance as the standard finite element analysis on fitted meshes. • Full decoupling of phononic crystal geometry and mesh without loss of accuracy. • The unit cell boundaries and inclusion geometries can be varied without remeshing. • Bloch–Floquet BCs are prescribed strongly on edges that do not match the mesh. • Transforms non-conforming periodicity into an enriched node-to-node periodicity. • Also suitable for standard and Bloch–Floquet periodic BCs on non-periodic meshes. [ABSTRACT FROM AUTHOR]

We consider solving the surface Helmholtz equation on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions. Using a stabilized method combining Galerkin least squares stabilization and a penalty on the gradient jumps we obtain stability of the discrete formulation under the condition h k < C , where h denotes the mesh size, k the wave number and C a constant depending mainly on the surface curvature κ , but not on the surface/mesh intersection. Optimal error estimates in the H 1 and L 2 -norms follow. • Implementation and analysis of a TraceFEM for the Helmholtz equation on surfaces. • Stablizing terms make the semi-discretized method absolutely stable. • The fully discrete method is stable under a mild mesh condition h k < C. • Optimal error bounds in the H 1 - and L 2 -norms. • The numerical performance compared to the standard Galerkin method is excellent in situations close to resonance. [ABSTRACT FROM AUTHOR]