Higher order triangular basis functions and solution performance of the CG method

Abdul-Rahman, Razi; Kasper, Manfred

doi:10.1016/j.cma.2007.07.014

Cited by 5 publications

(4 citation statements)

References 15 publications

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“…A new class of higher-order finite elements based on generalized eigenfunctions of the Laplace operator was presented in [7]. In [11], a set of hierarchical high-order basis functions for triangles was constructed using a systematic orthogonalization approach that yields better conditioning. High-order bases have been also developed for the mixed finite element methods as in [12,13].…”

Section: Introductionmentioning

confidence: 99%

“…A hybrid preconditioning scheme employing a nonorthogonal basis that combines global and locally accelerated preconditioners for rapid iterative diagonalization of generalized eigenvalue problems in electronic structure calculations was proposed in [23]. Numerical preconditioners have been also developed for other numerical methods in recent years, as can be seen in [24,25,26,11,27]. In [24] was established a scaling relation between the condition number of the system matrices and the smallest cell volume fraction for the Finite Cell Method.…”

Section: Introductionmentioning

confidence: 99%

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Simulation of non-linear structural elastodynamic and impact problems using minimum energy and simultaneous diagonalization high-order bases

Dias,

Suzuki,

Valente

et al. 2021

Preprint

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We present the application of simultaneous diagonalization and minimum energy (SDME) high-order finite element modal bases for simulation of transient non-linear elastodynamic problem, including impact cases with neohookean hyperelastic materials. The bases are constructed using procedures for simultaneous diagonalization of the internal modes and Schur complement of the boundary modes from the standard nodal and modal bases, constructed using Lagrange and Jacobi polynomials, respectively. The implementation of these bases in a high-order finite element code is straightforward, since the procedure is applied only to the one-dimensional expansion bases. Non-linear transient structural problems with large deformation, hyperelastic materials and impact are solved using the obtained bases with explicit and implicit time integration procedures. Iterative solutions based on preconditioned conjugate gradient methods are considered. The performance of the proposed bases in terms of the number of iterations of pre-conditioned conjugate gradient methods and computational time are compared with the standard nodal and modal bases. Our numerical tests obtained speedups up to 41 using the considered bases when compared to the standard ones.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Simulation of non-linear structural elastodynamic and impact problems using minimum energy and simultaneous diagonalization high-order bases

Dias,

Suzuki,

Valente

et al. 2021

Preprint

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“…Among all these choices, the Jacobi polynomials have been employed in the construction of several recent expansion bases , and have been proved to be very successful in computational fluid dynamics. From the numerical standpoint, the spectral/hp element version based on general Jacobi polynomials gives a good sparsity and is well conditioned for enhanced computational efficiency and scalability as shown in . It also provides further flexibility in generating polymorphic elements, for example, hexahedral, tetrahedral, triangular prisms and even pyramids.…”

Section: Introductionmentioning

confidence: 99%

“…where O F n;k 1 is defined in (18), 4U is the vector of the expansion coefficients .4 O u e / ip and M is the global mass matrix organized from the element-wise matrices (8). R n;k 1 is part of the residual vector with length 3N , which can be assembled from local vectors with components…”

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confidence: 99%

A semi‐local spectral/hp element solver for linear elasticity problems

Bittencourt

Karniadakis

2014

Numerical Meth Engineering

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SUMMARYWe develop an efficient semi‐local method for speeding up the solution of linear systems arising in spectral/hp element discretization of the linear elasticity equations. The main idea is to approximate the element‐wise residual distribution with a localization operator we introduce in this paper, and subsequently solve the local linear system. Additionally, we decouple the three directions of displacement in the localization operator, hence enabling the use of an efficient low energy preconditioner for the conjugate gradient solver. This approach is effective for both nodal and modal bases in the spectral/hp element method, but here, we focus on the modal hierarchical basis. In numerical tests, we verify that there is no loss of accuracy in the semi‐local method, and we obtain good parallel scalability and substantial speed‐up compared to the original formulation. In particular, our tests include both structure‐only and fluid‐structure interaction problems, with the latter modeling a 3D patient‐specific brain aneurysm. Copyright © 2014 John Wiley & Sons, Ltd.

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Simulation of non-linear structural elastodynamic and impact problems using minimum energy high-order bases

Dias

Suzuki²,

Valente

et al. 2022

International Journal of Mechanical Sciences

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Higher order triangular basis functions and solution performance of the CG method

Cited by 5 publications

References 15 publications

Simulation of non-linear structural elastodynamic and impact problems using minimum energy and simultaneous diagonalization high-order bases

Simulation of non-linear structural elastodynamic and impact problems using minimum energy and simultaneous diagonalization high-order bases

A semi‐local spectral/hp element solver for linear elasticity problems

Simulation of non-linear structural elastodynamic and impact problems using minimum energy high-order bases

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