| Date: | October 13, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Frederic Marazzato, LSU |
| Title: | Variational discrete element methods |
| Abstract: | Discrete Element Methods (DEM) have been introduced in [Hoover et al, 1974] to compute granular materials. Their application to compute elastic materials has remained an open question for a long time [Jebahi et al, 2015]. A first step in that direction was achieved in [Monasse et al, 2012], however the method suffered from several limitations. In [Marazzato et al, 2020], a discretization method for dynamic elasto-plasticity was proposed based on DEM by making a link with hybrid finite volume methods. Only cell dofs are used and a reconstruction is devised to obtain P^1 non-conforming polynomials in each cell and thus constant strains and stresses in each cell. An adaptation of the method consisting in adding cellwise constant rotational dofs made possible the computation of Cosserat materials [Marazzato, 2021]. Taking advantage of the capacity of DEM to deal with discontinuous displacement fields, another adaptation of the method made possible the computation of fracture in two-dimensional settings. Numerical examples for both static and dynamic computations in two and three dimensions will demonstrate the robustness of the proposed methodology. |
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| Date: | October 27, 2021 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Giselle Sosa Jones, University of Houston |
| Title: | A space-time hybridizable discontinuous Galerkin method for free-surface waves |
| Abstract: | The numerical modeling of free-surface waves requires the solution of a system of partial differential equations that govern the movement of the fluid, coupled with free-surface boundary conditions that describe the free-surface. In these problems, the domain where the partial differential equations are posed depends on the free-surface which leads to a strong coupling between the flow equations and the free-surface. In order to effectively track the free-surface, we require an accurate discretization technique that can handle moving domains. Moreover, it is desirable for the discretization to handle the mesh movement without postprocessing techniques. In this talk, we present a space-time hybridizable discontinuous Galerkin discretization of linear and nonlinear free-surface wave problems. We obtain a priori error estimates for the linearized problem, and present various numerical examples that demonstrate the capabilities of the method. |
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| Date: | November 10, 2021 |
| Time: | 1:00pm |
| Location: | Zoom |
| Speaker: | Maryam Parvizi, Leibniz University of Hannover |
| Title: | On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion |
| Abstract: | In this talk, we consider locally L2(Ω)-stable operators mapping into spaces of continuous piecewise
polynomial set on shape regular meshes with certain approximation properties in L2(Ω). For such
operators, we discuss the following stability results:
•These operators are stable mappings H3/2(Ω) →B3/2
2,∞(Ω).
•If the mesh is additionally quasi-uniform, for the space of continuous piecewise polynomials on this
mesh, we have a sharper stability estimate B3/2
2,∞(Ω) →B3/2
2,∞(Ω).
Given a mesh T obtained by Newest Vertex Bisection (NVB) refinement from a regular triangulation
̂T0 and ̂T` as the sequence of uniformly refined NVB-generated meshes, we introduce the finest common
coarsening (fcc) of two meshes ̃T` := fcc(T, ̂T`). For the space of continuous piecewise polynomials
defined on the mesh hierarchy ̃T`, we construct the modified Scott-Zhang operator ̃ISZ` in such a way
that for continuous piecewise polynomials on T, this operator coincides with the Scott-Zhang operator̂ISZ` on ̂T`.
Since the Scott-Zhang operators are local, L2(Ω)-stable operators with certain approximation prop-
erties in L2(Ω), therefore these operators admit the above stability results. Taking advantage of the
stability results and the mentioned property of the modified Scott-Zhang operators, we present multi-
level norm equivalences in the Besov spaces B3θ/2
2,q (Ω), θ ∈(0,1), q ∈[1,∞].
As an application, we present a local multilevel diagonal preconditioner for the integral fractional
Laplacian (−∆)s for s ∈ (0,1) on adaptively refined meshes and prove this multilevel diagonal scaling
gives rise to uniformly bounded condition number for the integral fractional Laplacian. To prove the
main result, we apply the norm equivalence of the multilevel decomposition. |
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| Date: | November 17, 2021 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | William Pazner, LLNL |
| Title: | Low-order methods for high-order finite element discretizations and solvers |
| Abstract: | In this talk, I will discuss two applications of low-order methods to increase the efficiency and robustness of high-order finite element discretizations and solvers. In the first part of the talk, I will discuss matrix-free linear solvers for high-order discontinuous Galerkin discretizations of elliptic problems. These solvers are based on the spectral equivalence of the high-order discretization with a low-order refined discretization (often known as the "FEM-SEM equivalence"). A novel extension of this equivalence to the case of nonconforming meshes and variable polynomial degrees will be presented. Using the subspace correction (additive Schwarz) framework, robust preconditioners for DG discretizations with (nonconforming) hp-refinement will be constructed that result in uniform convergence with respect to mesh size, polynomial degree, and DG penalty parameter. This method is amenable for use on adaptively refined meshes with any degree of irregularity. Examples are shown using the interior penalty and BR2 methods. In the second part of the talk, I will discuss the construction of invariant domain preserving discontinuous Galerkin methods using subcell convex limiting (cf. Guermond and Popov, 2016). The high-order DG method is augmented with a sparse low-order Lax-Friedrichs discretization constructed on a refined mesh. A key feature is that the low-order method does not become more dissipative as the polynomial degree of the high-order method is increased, in contrast with other graph viscosity techniques. The high-order and low-order methods are blended using an efficient dimension-by-dimension convex limiting procedure that can be used to guarantee the preservation of any number of user-specified convex invariants while retaining subcell resolution. Several numerical examples for the Euler equations will be shown, for which this method preserves the positivity of density, pressure and internal energy, and satisfies a minimum principle for the specific entropy. |
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| Date: | December 1, 2021 |
| Time: | 1:00pm |
| Location: | zoom |
| Speaker: | Amirreza Khodaddian, Leibniz University of Hannover |
| Title: | Uncertainty quantification in phase-field fracture problem |
| Abstract: | Phase-field fracture is a very active research field with numerous applications. The model is used to describe the crack propagation in brittle materials (e.g., concrete and ceramics), ductile materials such as metals and steel, and hydraulic fracture (extracting oil and natural gases). Moreover, the uncertainty arises from the heterogeneity of the material structure and spatial fluctuation of the material properties. The challenging part is the multiphysics fracture framework since we should deal with different subdomains (each has different PDEs), which significantly increases the computational costs. In brittle fracture, the computational model is based on the coupling of the elasticity equation (to model displacement) and the phase-field fracture (modeling the crack propagation). In hydraulic fracture, these equations are additionally coupled with the Darcy-type flow to model fluid pressure. Here, mechanical and geo-mechanical parameters have an influential effect on the model simulations; however, most of these parameters can not be estimated experimentally. Bayesian inversion is an efficient and reliable probabilistic model to estimate the material parameters when a synthetic/measured reference observation is available. In this talk, for phase-field fracture models, we employ Markov chain Monte Carlo (MCMC) techniques to estimate the posterior distributions of the parameters. In Bayesian inversion, hundreds or thousands of (PDE-based) forward runs are necessary. The simulations are computationally expensive since, in order to achieve reliable accuracy, a significant degree of freedom is needed. Next, we develop multiscale techniques, specifically non-intrusive global/local approach in which a fine-scale problem is solved in the fracture region and a linearized coarse problem in the remaining domain. The global/local setting is coupled with Bayesian inversion to identify the material parameters and model the crack pattern. The results show a significant computational cost reduction compared to the full mode |
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