A new adaptive refinement strategy for phasefield models of brittle fracture is proposed. The approach provides a computationally efficient solution to the high demand in spatial resolution of phase-field models. The strategy is based on considering two types of elements: h-refined elements along cracks, where more accuracy is needed to capture the solution, and standard elements in the rest of the domain. Continuity between adjacent elements of different type is imposed in weak form by means of Nitsche's method. The weakly imposition of continuity leads to a very local refinement in a simple way, for any degree of approximation and both in 2D and 3D. The performance of the strategy is assessed for several scenarios in the quasi-static regime, including coalescence and branching of cracks in 2D and a twisting crack in 3D.
Summary In this paper, we propose an adaptive refinement strategy for phase‐field models of brittle fracture, which is based on a novel hybridizable discontinuous Galerkin (HDG) formulation of the problem. The adaptive procedure considers standard elements and only one type of h‐refined elements, dynamically located along the propagating cracks. Thanks to the weak imposition of interelement continuity in HDG methods, and in contrast with other existing adaptive approaches, hanging nodes or special transition elements are not needed, which simplifies the implementation. Various numerical experiments, including one branching test, show the accuracy, robustness, and applicability of the presented approach to quasistatic phase‐field simulations.
Reduced-order models are essential tools to deal with parametric problems in the context of optimization, uncertainty quantification, or control and inverse problems. The set of parametric solutions lies in a low-dimensional manifold (with dimension equal to the number of independent parameters) embedded in a large-dimensional space (dimension equal to the number of degrees of freedom of the full-order discrete model). A posteriori model reduction is based on constructing a basis from a family of snapshots (solutions of the full-order model computed offline), and then use this new basis to solve the subsequent instances online. Proper orthogonal decomposition (POD) reduces the problem into a linear subspace of lower dimension, eliminating redundancies in the family of snapshots. The strategy proposed here is to use a nonlinear dimensionality reduction technique, namely, the kernel principal component analysis (kPCA), in order to find a nonlinear manifold, with an expected much lower dimension, and to solve the problem in this low-dimensional manifold. Guided by this paradigm, the methodology devised here introduces different novel ideas, namely, 1) characterizing the nonlinear manifold using local tangent spaces, where the reduced-order problem is linear and based on the neighboring snapshots, 2) the approximation space is enriched with the cross-products of the snapshots, introducing a quadratic description, 3) the kernel for kPCA is defined ad hoc, based on physical considerations, and 4) the iterations in the reduced-dimensional space are performed using an algorithm based on a Delaunay tessellation of the cloud of snapshots in the reduced space. The resulting computational strategy is performing outstandingly in the numerical tests, alleviating many of the problems associated with POD and improving the numerical accuracy.
Scaffolds are microporous biocompatible structures that serve as material support for cells to proliferate, differentiate and form functional tissue. In particular, in the field of bone regeneration, insertion of scaffolds in a proper physiological environment is known to favour bone formation by releasing calcium ions, among others, triggering differentiation of mesenchymal cells into osteoblasts. Computational simulation of molecular distributions through scaffolds is a potential tool to study the scaffolds’ performance or optimal designs, to analyse their impact on cell differentiation, and also to move towards reduction in animal experimentation. Unfortunately, the required numerical models are often highly complex and computationally too costly to develop parametric studies. In this context, we propose a computational parametric reduced-order model to obtain the distribution of calcium ions in the interstitial fluid flowing through scaffolds, depending on several physical parameters. We use the well-known Proper Orthogonal Decomposition (POD) with two different variations: local POD and POD with quadratic approximations. Computations are performed using two realistic geometries based on a foamed and a 3D-printed scaffolds. The location of regions with high concentration of calcium in the numerical simulations is in fair agreement with regions of bone formation shown in experimental observations reported in the literature. Besides, reduced-order solutions accurately approximate the reference finite element solutions, with a significant decrease in the number of degrees of freedom, thus avoiding computationally expensive simulations, especially when performing a parametric analysis. The proposed reduced-order model is a competitive tool to assist the design of scaffolds in osteoinduction research.
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