54 research outputs found
Mixed aggregated finite element methods for the unfitted discretization of the Stokes problem
In this work, we consider unfitted finite element methods for the numerical approximation of the Stokes problem. It is well-known that these kinds of methods lead to arbitrarily ill-conditioned systems and poorly approximated fluxes on unfitted interfaces/boundaries. In order to solve these issues, we consider the recently proposed aggregated finite element method, originally motivated for coercive problems. However, the well-posedness of the Stokes problem is far more subtle and relies on a discrete inf-sup condition. We consider mixed finite element methods that satisfy the discrete version of the inf-sup condition for body-fitted meshes and analyze how the discrete inf-sup is affected when considering the unfitted case. We propose different aggregated mixed finite element spaces combined with simple stabilization terms, which can include pressure jumps and/or cell residuals, to fix the potential deficiencies of the aggregated inf-sup. We carry out a complete numerical analysis, which includes stability, optimal a priori error estimates, and condition number bounds that are not affected by the small cut cell problem. For the sake of conciseness, we have restricted the analysis to hexahedral meshes and discontinuous pressure spaces. A thorough numerical experimentation bears out the numerical analysis. The aggregated mixed finite element method is ultimately applied to two problems with nontrivial geometries. No separate or additional fees are collected for access to or distribution of the work
Analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem
In this paper we present a new H(div)-conforming unfitted finite element
method for the mixed Poisson problem which is robust in the cut configuration
and preserves conservation properties of body-fitted finite element methods.
The key is to formulate the divergence-constraint on the active mesh, instead
of the physical domain, in order to obtain robustness with respect to cut
configurations without the need for a stabilization that pollutes the mass
balance. This change in the formulation results in a slight inconsistency, but
does not affect the accuracy of the flux variable. By applying post-processings
for the scalar variable, in virtue of classical local post-processings in
body-fitted methods, we retain optimal convergence rates for both variables and
even the superconvergence after post-processing of the scalar variable. We
present the method and perform a rigorous a-priori error analysis of the method
and discuss several variants and extensions. Numerical experiments confirm the
theoretical results.Comment: 26 pages, 6 figure
Neural Level Set Topology Optimization Using Unfitted Finite Elements
To facilitate widespread adoption of automated engineering design techniques,
existing methods must become more efficient and generalizable. In the field of
topology optimization, this requires the coupling of modern optimization
methods with solvers capable of handling arbitrary problems. In this work, a
topology optimization method for general multiphysics problems is presented. We
leverage a convolutional neural parameterization of a level set for a
description of the geometry and use this in an unfitted finite element method
that is differentiable with respect to the level set everywhere in the domain.
We construct the parameter to objective map in such a way that the gradient can
be computed entirely by automatic differentiation at roughly the cost of an
objective function evaluation. The method produces optimized topologies that
are similar in performance yet exhibit greater regularity than baseline
approaches on standard benchmarks whilst having the ability to solve a more
general class of problems, e.g., interface-coupled multiphysics.Comment: 16 pages + refs, 10 fig
A Quasi-Conforming Embedded Reproducing Kernel Particle Method for Heterogeneous Materials
We present a quasi-conforming embedded reproducing kernel particle method
(QCE-RKPM) for modeling heterogeneous materials that makes use of techniques
not available to mesh-based methods such as the finite element method (FEM) and
avoids many of the drawbacks in current embedded and immersed formulations
which are based on meshed methods. The different material domains are
discretized independently thus avoiding time-consuming, conformal meshing. In
this approach, the superposition of foreground (inclusion) and background
(matrix) domain integration smoothing cells are corrected by a quasi-conforming
quadtree subdivision on the background integration smoothing cells. Due to the
non-conforming nature of the background integration smoothing cells near the
material interfaces, a variationally consistent (VC) correction for domain
integration is introduced to restore integration constraints and thus optimal
convergence rates at a minor computational cost. Additional interface
integration smoothing cells with area (volume) correction, while
non-conforming, can be easily introduced to further enhance the accuracy and
stability of the Galerkin solution using VC integration on non-conforming
cells. To properly approximate the weak discontinuity across the material
interface by a penalty-free Nitsche's method with enhanced coercivity, the
interface nodes on the surface of the foreground discretization are also shared
with the background discretization. As such, there are no tunable parameters,
such as those involved in the penalty type method, to enforce interface
compatibility in this approach. The advantage of this meshfree formulation is
that it avoids many of the instabilities in mesh-based immersed and embedded
methods. The effectiveness of QCE-RKPM is illustrated with several examples
Residual-based error estimation and adaptivity for stabilized immersed isogeometric analysis using truncated hierarchical B-splines
We propose an adaptive mesh refinement strategy for immersed isogeometric
analysis, with application to steady heat conduction and viscous flow problems.
The proposed strategy is based on residual-based error estimation, which has
been tailored to the immersed setting by the incorporation of appropriately
scaled stabilization and boundary terms. Element-wise error indicators are
elaborated for the Laplace and Stokes problems, and a THB-spline-based local
mesh refinement strategy is proposed. The error estimation .and adaptivity
procedure is applied to a series of benchmark problems, demonstrating the
suitability of the technique for a range of smooth and non-smooth problems. The
adaptivity strategy is also integrated in a scan-based analysis workflow,
capable of generating reliable, error-controlled, results from scan data,
without the need for extensive user interactions or interventions.Comment: Submitted to Journal of Mechanic
Embedded multilevel monte carlo for uncertainty quantification in random domains
The multilevel Monte Carlo (MLMC) method has proven to be an effective variance-reduction statistical method for uncertainty quantification (UQ) in partial differential equation (PDE) models. It combines approximations at different levels of accuracy using a hierarchy of meshes whose generation is only possible for simple geometries. On top of that, MLMC and Monte Carlo (MC) for random domains involve the generation of a mesh for every sample. Here we consider the use of embedded methods which make use of simple background meshes of an artificial domain (a bounding-box) for which it is easy to define a mesh hierarchy. We use the recent aggregated finite element method (AgFEM) method, which permits to avoid ill-conditioning due to small cuts, to design an embedded MLMC (EMLMC) framework for (geometrically and topologically) random domains implicitly defined through a random level-set function. Predictions from existing theory are verified in numerical experiments and the use of AgFEM is statistically demonstrated to be crucial for complex and uncertain geometries in terms of robustness and computational cost.Peer ReviewedPostprint (author's final draft
Embedded multilevel Monte Carlo for uncertainty quantification in random domains
The multilevel Monte Carlo (MLMC) method has proven to be an effective variance-reduction statistical method for Uncertainty quantification in PDE models. It combines approximations at different levels of accuracy using a hierarchy of meshes in a similar way as multigrid. The generation of body-fitted mesh hierarchies is only possible for simple geometries. On top of that, MLMC for random domains involves the generation of a mesh for every sample. Instead, here we consider the use of embedded methods which make use of simple background meshes of an artificial domain (a bounding-box) for which it is easy to define a mesh hierarchy, thus eliminating the need of body-fitted unstructured meshes, but can produce ill-conditioned discrete problems. To avoid this complication, we consider the recent aggregated finite element method (AgFEM). In particular, we design an embedded MLMC framework for (geometrically and topologically) random domains implicitly defined through a random level-set function, which makes use of a set of hierarchical background meshes and the AgFEM. Performance predictions from existing theory are verified statistically in three numerical experiments, namely the solution of the Poisson equation on a circular domain of random radius, the solution of the Poisson equation on a topologically identical but more complex domain, and the solution of a heat-transfer problem in a domain that has geometric and topological uncertainties. Finally, the use of AgFE is statistically demonstrated to be crucial for complex and uncertain geometries in terms of robustness and computational cost. Date: November 28, 2019
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