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Results for “mixed finite element methods”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2015

Postprocessing Mixed Finite Element Methods For Solving Cahn–Hilliard Equation: Methods and Error Analysis

A postprocessing technique for mixed finite element methods for the Cahn-Hilliard equation is developed and analyzed. Once the mixed finite element approximations have been computed at a fixed time on the coarser mesh, the approximations are postprocessed by solving two decoupled Poisson equations in an enriched finite element space (either on a finer grid or a higher-order space) for which many fast Poisson solvers can be applied. The nonlinear iteration is only applied to a much smaller size problem and the computational cost using Newton and direct solvers is negligible compared with the cost of the linear problem. The analysis presented here shows that this technique remains the optimal rate of convergence for both the concentration and the chemical potential approximations. The corresponding error estimate obtained in our paper, especially the negative norm error estimates, are non-trivial and different with the existing results in the literatures.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

A coupled multipoint stress–multipoint flux mixed finite element method for the Biot system of poroelasticity

In this work, we present a mixed finite element method for a five-field formulation of the Biot system of poroelasticity that reduces to a cell-centered pressure–displacement system on simplicial and quadrilateral grids. A mixed stress–displacement–rotation formulation for elasticity with weak stress symmetry is coupled with a mixed velocity–pressure Darcy formulation. The spatial discretization is based on combining the multipoint stress mixed finite element (MSMFE) method for elasticity and the multipoint flux mixed finite element (MFMFE) method for Darcy flow. It uses the lowest order Brezzi–Douglas–Marini mixed finite element spaces for the poroelastic stress and Darcy velocity, piecewise constant displacement and pressure, and continuous piecewise linear or bilinear rotation. A vertex quadrature rule is applied to the velocity, stress, and stress–rotation bilinear forms, which block-diagonalizes the corresponding matrices and allows for local velocity, stress, and rotation elimination. This leads to a cell-centered positive-definite system for pressure and displacement at each time step. We perform error analysis for the semidiscrete and fully discrete formulations, establishing first order convergence for all variables in their natural norms. The numerical tests confirm the theoretical convergence rates and illustrate the locking-free property of the method.

42 ENGINEERING↗
DOE OSTI2019

Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra

We develop higher order multipoint flux mixed finite element (MFMFE) methods for solving elliptic problems on quadrilateral and hexahedral grids that reduce to cell-based pressure systems. The methods are based on a new family of mixed finite elements, which are enhanced Raviart–Thomas spaces with bubbles that are curls of specially chosen polynomials. The velocity degrees of freedom of the new spaces can be associated with the points of tensor-product Gauss–Lobatto quadrature rules, which allows for local velocity elimination and leads to a symmetric and positive definite cell-based system for the pressures. We prove optimal [Formula: see text]th order convergence for the velocity and pressure in their natural norms, as well as [Formula: see text]st order superconvergence for the pressure at the Gauss points. Moreover, local postprocessing gives a pressure that is superconvergent of order [Formula: see text] in the full [Formula: see text]-norm. Numerical results illustrating the validity of our theoretical results are included.

Mathematics↗
DOE OSTI2019

Domain decomposition and multiscale mortar mixed finite element methods for linear elasticity with weak stress symmetry

Two non-overlapping domain decomposition methods are presented for the mixed finite element formulation of linear elasticity with weakly enforced stress symmetry. The methods utilize either displacement or normal stress Lagrange multiplier to impose interface continuity of normal stress or displacement, respectively. By eliminating the interior subdomain variables, the global problem is reduced to an interface problem, which is then solved by an iterative procedure. The condition number of the resulting algebraic interface problem is analyzed for both methods. A multiscale mortar mixed finite element method for the problem of interest on non-matching multiblock grids is also studied. It uses a coarse scale mortar finite element space on the non-matching interfaces to approximate the trace of the displacement and impose weakly the continuity of normal stress. A priori error analysis is performed. It is shown that, with appropriate choice of the mortar space, optimal convergence on the fine scale is obtained for the stress, displacement, and rotation, as well as some superconvergence for the displacement. Computational results are presented in confirmation of the theory of all proposed methods.

Mathematics↗
DOE OSTI2024

Generalized Korn’s inequalities for piecewise $H¹$ and $H²$ vector fields

The purpose of this paper is to construct a new class of discrete generalized Korn’s inequalities for piecewise $H^1$ vector fields and piecewise $H^2$ vector fields in three-dimensional space. The resulting Korn’s inequalities are different from the standard Korn’s inequalities, as they involve the trace-free symmetric gradient operator, in place of the usual symmetric gradient operator. Furthermore, it is anticipated that the new generalized Korn’s inequalities will be useful for the analysis of a broad range of finite element methods, including mixed finite element methods and discontinuous Galerkin methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

A fourth-order phase-field fracture model: Formulation and numerical solution using a continuous/discontinuous Galerkin method

Modeling crack initiation and propagation in brittle materials is of great importance to be able to predict sudden loss of load-carrying capacity and prevent catastrophic failure under severe dynamic loading conditions. Second-order phase-field fracture models have gained wide adoption given their ability to capture the formation of complex fracture patterns, e.g. via crack merging and branching, and their suitability for implementation within the context of the conventional finite element method. Higher-order phase-field models have also been proposed to increase the regularity of the exact solution and thus increase the spatial convergence rate of its numerical approximation. However, they require special numerical techniques to enforce the necessary continuity of the phase field solution. In this paper, we derive a fourth-order phase-field model of fracture in two independent ways; namely, from Hamilton’s principle and from a higher-order micromechanics-based approach. The latter approach is novel, and provides a physical interpretation of the higher-order terms in the model. In addition, we propose a continuous/discontinuous Galerkin (C/DG) method for use in computing the approximate phase-field solution. This method employs Lagrange polynomial shape functions to guarantee -continuity of the solution at inter-element boundaries, and enforces the required regularity with the aid of additional variational and interior penalty terms in the weak form. Finally, the phase-field equation is coupled with the momentum balance equation to model dynamic fracture problems in hyper-elastic materials. Two benchmark problems are presented to compare the numerical behavior of the C/DG method with mixed finite element methods.

42 ENGINEERING↗
DOE OSTI2018

Mixed finite-element method for multi-term time-fractional diffusion and diffusion-wave equations

In this paper, numerical theory based on the mixed finite-element method and finite difference analog of the Caputo fractional derivative for multi-term time-fractional diffusion equations and diffusion-wave equations is analyzed. The unconditional stability and convergence results are proved for the two resulting fully discrete schemes. Finally, the obtained results are supported by numerical experiments carried out for some test problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Generalization of mixed multiscale finite element methods with applications

Many science and engineering problems exhibit scale disparity and high contrast. The small scale features cannot be omitted in the physical models because they can affect the macroscopic behavior of the problems. However, resolving all the scales in these problems can be prohibitively expensive. As a consequence, some types of model reduction techniques are required to design efficient solution algorithms. For practical purpose, we are interested in mixed finite element problems as they produce solutions with certain conservative properties. Existing multiscale methods for such problems include the mixed multiscale finite element methods. We show that for complicated problems, the mixed multiscale finite element methods may not be able to produce reliable approximations. This motivates the need of enrichment for coarse spaces. Two enrichment approaches are proposed, one is based on generalized multiscale finte element metthods (GMsFEM), while the other is based on spectral element-based algebraic multigrid (rAMGe). The former one, which is called mixed GMsFEM, is developed for both Darcy’s flow and linear elasticity. Application of the algorithm in two-phase flow simulations are demonstrated. For linear elasticity, the algorithm is subtly modified due to the symmetry requirement of the stress tensor. The latter enrichment approach is based on rAMGe. The algorithm differs from GMsFEM in that both of the velocity and pressure spaces are coarsened. Due the multigrid nature of the algorithm, recursive application is available, which results in an efficient multilevel construction of the coarse spaces. Stability, convergence analysis, and exhaustive numerical experiments are carried out to validate the proposed enrichment approaches. iii

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Physics-preserving enriched Galerkin method for a fully-coupled thermo-poroelasticity model

This paper proposes a new numerical method for a fully-coupled, quasi-static thermo-poroelasticity model in a unified enriched Galerkin (EG) method framework. In our method, the mechanics sub-problem is solved using a locking-free EG method, and the flow and heat sub-problems are solved using a locally-conservative EG method. The proposed method offers mass and energy conservation properties with much lower costs than other methods with the same properties, including discontinuous Galerkin methods and mixed finite element methods. The well-posedness and optimal a priori error estimates are carefully derived. Here, several numerical tests confirm the theoretical optimal convergence rates and the mass and energy conservation properties of the new method.

15 GEOTHERMAL ENERGY↗
DOE OSTI2016

Mimetic Finite Differences and Algebraic Multilevel Methods for Reservoir Performance Predictions: CRADA Final Report

The need to better predict and control reservoir performance is driving the development of increasingly sophisticated reservoir models that use unstructured general polyhedral grids to capture highly detailed stratigraphy. However, both discretization methods and scalable iterative solution algorithms for these situations are an active area of research. In this work we compared the performance of a new Mimetic Finite Difference (MFD) method with two new Mixed Finite Element methods, and LANL and the CRADA Participant investigated the performance of Los Alamos National Laboratory’s algebraic multigrid code on realistic flow problems. Furthermore, uncertainty quantification requires rapid evaluations of reservoir models. To do this, fine-scale reservoir models need to be coarsened and yet important fine-scale features and processes that have a significant impact on flows need to be preserved. To this end, we investigated the use of multi-resolution methods, such as the Multilevel Multiscale Mimetic (M3) method, to coarsen reservoir models.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Nonlocal and Mixed-Locality Multiscale Finite Element Methods

In many applications the resolution of small-scale heterogeneities remains a significant hurdle to robust and reliable predictive simulations. In particular, while material variability at the mesoscale plays a fundamental role in processes such as material failure, the resolution required to capture mechanisms at this scale is often computationally intractable. Multiscale methods aim to overcome this difficulty through judicious choice of a subscale problem and a robust manner of passing information between scales. One promising approach is the multiscale finite element method, which increases the fidelity of macroscale simulations by solving lower-scale problems that produce enriched multiscale basis functions. Here, in this study, we present the first work toward application of the multiscale finite element method to the nonlocal peridynamic theory of solid mechanics. This is achieved within the context of a discontinuous Galerkin framework that facilitates the description of material discontinuities and does not assume the existence of spatial derivatives. Analysis of the resulting nonlocal multiscale finite element method is achieved using the ambulant Galerkin method, developed here with sufficient generality to allow for application to multiscale finite element methods for both local and nonlocal models that satisfy minimal assumptions. Finally, we conclude with preliminary results on a mixed-locality multiscale finite element method in which a nonlocal model is applied at the fine scale and a local model at the coarse scale.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

A high-order staggered meshless method for elliptic problems

Here, we present a new meshless method for scalar diffusion equations, which is motivated by their compatible discretizations on primal-dual grids. Unlike the latter though, our approach is truly meshless because it only requires the graph of nearby neighbor connectivity of the discretization points. This graph defines a local primal-dual grid complex with a virtual dual grid, in the sense that specification of the dual metric attributes is implicit in the method's construction. Our method combines a topological gradient operator on the local primal grid with a generalized moving least squares approximation of the divergence on the local dual grid. We show that the resulting approximation of the div-grad operator maintains polynomial reproduction to arbitrary orders and yields a meshless method, which attains $O(h^{m})$ convergence in both $L^2$- and $H^1$-norms, similar to mixed finite element methods. We demonstrate this convergence on curvilinear domains using manufactured solutions in two and three dimensions. Application of the new method to problems with discontinuous coefficients reveals solutions that are qualitatively similar to those of compatible mesh-based discretizations.

97 MATHEMATICS AND COMPUTING↗