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Results for “adaptive finite elements”

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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All sourcesNASA NTRSDOE OSTI
NASA NTRS1995

Auto-adaptive finite element meshes

Accurate capturing of discontinuities within compressible flow computations is achieved by coupling a suitable solver with an automatic adaptive mesh algorithm for unstructured triangular meshes. The mesh adaptation procedures developed rely on non-hierarchical dynamical local refinement/derefinement techniques, which hence enable structural optimization as well as geometrical optimization. The methods described are applied for a number of the ICASE test cases are particularly interesting for unsteady flow simulations.

Richter, Roland↗
NASA NTRS1978

Adaptive finite element analysis based on p-convergence

The results of numerical experiments are presented in which a posteriori estimators of error in strain energy were examined on the basis of a typical problem in linear elastic fracture mechanics. Two estimators were found to give close upper and lower bounds for the strain energy error. The potential significance of this is that the same estimators may provide a suitable basis for adaptive redistribution of the degrees of freedom in finite element models.

Szabo, B. A.↗
NASA NTRS1988

Paradigmatic error calculations for adaptive finite element approximations of convection dominated flows

New error estimation techniques are described for two-dimensional convection-dominated flows. Examples of applications to linear convection problems are given. The methods are based on discontinuous finite-element techniques. Results show that conventional interpolation methods may lead to gross underestimates of actual error and be inadequate for use in adaptive methods.

Oden, J. T.↗
NASA NTRS1987

An adaptive finite element method for high speed flows

The solution of the equations of compressible high speed flow, on unstructured triangular grids in 2D and tetrahedral grids in 3D, is considered. Solution methods based upon both Taylor-Galerkin and Runge-Kutta time-stepping techniques are presented and the incorporation of the ideas of flux corrected transport (FCT) is discussed. These methods are combined with an adaptive mesh regeneration procedure and are employed in the solution of several examples, consisting of Euler flows in both 2D and 3D and Navier-Stokes flows in 2D.

Peraire, J.↗
NASA NTRS1987

An adaptive finite element scheme for the Euler and Navier-Stokes equations

The implementation of two explicit finite element schemes for the solution of the compressible Euler and Navier-Stokes equations is presented. The schemes can be employed with general unstructured triangular meshes in two dimensions. Either scheme can therefore be used as the basic solver in a solution adaptive mesh procedure in a direct manner. The particular adaptive approach which is advocated here is intended for the solution of steady state problems only and involves an adaptive regeneration of the grid at prescribed stages during the false transient. The grid regeneration is accomplished by a mesh generator which has the capability of generating triangular grids over computational domains of arbitrary shape. The procedure is illustrated by solving transonic flows over multi-airfoil configurations and high speed flows, involving shock interactions, past circular cylinders.

Morgan, K.↗
NASA NTRS1985

An adaptive finite element procedure for compressible high speed flows

A practical finite element-based solution procedure for high-speed inviscid compressible flow problems is described. The method provides time-accurate solutions to the coompressible Euler equations, and is computationally more efficient than the one-step Taylor-Galerkin approach and better suited for implementation on the modern generation of vector computers. The method is coupled to an adaptive mesh refinement process that enables steady state solutions of improved quality to be obtained.

Loehner, R.↗
NASA NTRS1994

Parallel, adaptive finite element methods for conservation laws

We construct parallel finite element methods for the solution of hyperbolic conservation laws in one and two dimensions. Spatial discretization is performed by a discontinuous Galerkin finite element method using a basis of piecewise Legendre polynomials. Temporal discretization utilizes a Runge-Kutta method. Dissipative fluxes and projection limiting prevent oscillations near solution discontinuities. A posteriori estimates of spatial errors are obtained by a p-refinement technique using superconvergence at Radau points. The resulting method is of high order and may be parallelized efficiently on MIMD computers. We compare results using different limiting schemes and demonstrate parallel efficiency through computations on an NCUBE/2 hypercube. We also present results using adaptive h- and p-refinement to reduce the computational cost of the method.

Biswas, Rupak↗
NASA NTRS1979

Multi-level adaptive finite element methods. 1: Variation problems

A general numerical strategy for solving partial differential equations and other functional problems by cycling between coarser and finer levels of discretization is described. Optimal discretization schemes are provided together with very fast general solvers. It is described in terms of finite element discretizations of general nonlinear minimization problems. The basic processes (relaxation sweeps, fine-grid-to-coarse-grid transfers of residuals, coarse-to-fine interpolations of corrections) are directly and naturally determined by the objective functional and the sequence of approximation spaces. The natural processes, however, are not always optimal. Concrete examples are given and some new techniques are reviewed. Including the local truncation extrapolation and a multilevel procedure for inexpensively solving chains of many boundary value problems, such as those arising in the solution of time-dependent problems.

Brandt, A.↗
DOE OSTI2016

Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

Summary The finite element methods (FEMs) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. In this paper, we introduce the Adaptive Extended Stencil Finite Element Method (AES‐FEM) as a means for overcoming this dependence on element shape quality. Our method replaces the traditional basis functions with a set of generalized Lagrange polynomial basis functions , which we construct using local weighted least‐squares approximations. The method preserves the theoretical framework of FEM and allows imposing essential boundary conditions and integrating the stiffness matrix in the same way as the classical FEM. In addition, AES‐FEM can use higher‐degree polynomial basis functions than the classical FEM, while virtually preserving the sparsity pattern of the stiffness matrix. We describe the formulation and implementation of AES‐FEM and analyze its consistency and stability. We present numerical experiments in both 2D and 3D for the Poisson equation and a time‐independent convection–diffusion equation. The numerical results demonstrate that AES‐FEM is more accurate than linear FEM, is also more efficient than linear FEM in terms of error versus runtime, and enables much better stability and faster convergence of iterative solvers than linear FEM over poor‐quality meshes. Copyright © 2016 John Wiley & Sons, Ltd.

Conley, Rebecca↗
NASA NTRS1989

An adaptive implicit/explicit finite element scheme for compressible viscous high speed flow

An adaptive implicit/explicit finite element procedure for the solution of three-dimensional problems of steady compressible viscous high-speed flows is presented. In the vicinity of solid walls, a grid-exhibiting structure in the normal direction is employed, while away from this region the grid is totally unstructured. The implicit form of the algorithm is used near solid walls, with the grid structure being utilized in an equation solution approach based upon line relaxation. The explicit form of the algorithm is used elsewhere. Grid adaptation is achieved by means of adaptive remeshing. To illustrate the performance of the proposed method, solutions are obtained for the problems of shock-boundary layer interaction and shock-shock interaction on a swept cylindrical leading edge. Comparisons are made with experimental observations.

Hassan, O.↗
NASA NTRS1986

Recent advances in error estimation and adaptive improvement of finite element calculations

Recent advances in adaptive finite elements are summarized. General concepts behind a posteriori error estimation, h-method data management, and algorithms for fluid-mechanics applications are then examined, and some results of numerical experiments with new adaptive codes are given. Numerical examples include supersonic flow over a 20-deg ramp, supersonic flow in expansion corners, the reflecting-shock problem, and the rotating-cone problem. Finally, future directions of research in the field are outlined.

Oden, J. T.↗
NASA NTRS1995

Adaptive Finite-Element Computation In Fracture Mechanics

Report discusses recent progress in use of solution-adaptive finite-element computational methods to solve two-dimensional problems in linear elastic fracture mechanics. Method also shown extensible to three-dimensional problems.

Min, J. B.↗
NASA NTRS1998

Numerical Differentiation for Adaptively Refined Finite Element Meshes

Postprocessing of point-wise data is a fundamental process in many fields of research. Numerical differentiation is a key operation in computational electromagnetics. In the case of data obtained from a finite element method with automatic mesh refinement much work needs still to be done. This paper addresses some issues in differentiating data obtained from a finite element electromagnetic code with adaptive mesh refinement, and it proposes a methodology for deriving the electric field given the magnetic field on a mesh of linear triangular elements. The procedure itself is nevertheless more general and might be extended for numerically differentiating any point-wise solution based on triangular meshes.

Borgioli, Andrea↗
DOE OSTI2021

Automatically adaptive stabilized finite elements and continuation analysis for compaction banding in geomaterials

Under compressive creep, viscoplastic solids experiencing internal mass transfer processes can accommodate singular cnoidal wave solutions as material instabilities at the stationary wave limit. These instabilities appear when the loading rate is significantly faster than the material's capacity to diffusive internal perturbations, leading to localized failure features (e.g., cracks and compaction bands). These cnoidal waves, generally found in fluids, have strong nonlinearities that produce periodic patterns. Due to the singular nature of the solutions, the applicability of the theory is currently limited. Additionally, practical simulation tools require proper regularization to overcome the challenges that singularity induces. We focus on the numerical treatment of the governing equation using a nonlinear approach building on a recent adaptive stabilized finite element method. This automatic refinement method provides an error estimate that drives mesh adaptivity, a crucial feature for the problem at hand. We compare the performance of this adaptive strategy against analytical and standard finite element solutions. We then investigate the sensitivity of the diffusivity ratio, the parameter controlling the process, and identify multiple possible solutions with several stress peaks. We also show the evolution of the spacing between peaks for all solutions as a function of that parameter.

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