The solution of elastostatic and elastodynamic problems with Chebyshev spectral finite elements

Dauksher, Walter; Emery, A. F.

doi:10.1016/s0045-7825(99)00149-8

Cited by 58 publications

(32 citation statements)

References 23 publications

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“…(6) below) or by the modified integration rule technique (see Eqs. (7), (9) and (11) below). Inserting Eq.…”

Section: The Finite Element Techniques With Reduced Dispersion For Exmentioning

confidence: 97%

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Finite Element Modeling of Linear Elastodynamics Problems With Explicit Time-Integration Methods and Linear Elements With Reduced Dispersion. Comparative Study of Different Finite Element Techniques Used for Elastodynamics.

Idesman¹

2014

Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. We have developed two finite element techniques with reduced dispersion for linear elastodynamics that are used with explicit time-integration methods. These techniques are based on the modified integration rule for the mass and stiffness matrices and on the averaged mass matrix approaches that lead to the numerical dispersion reduction for linear finite elements. The analytical study of numerical dispersion for the new techniques is carried out in the 1-D, 2-D and 3-D cases. The numerical study of the effectiveness of the dispersion reduction techniques includes two-stage time-integration approach with the filtering stage (developed in our previous papers) that quantifies and removes spurious high-frequency oscillations from numerical results. We have found that in contrast to the standard linear elements with explicit time-integration methods and the lumped mass matrix, the finite element techniques with reduced dispersion yield more accurate results at small time increments (smaller than the stability limit) in the 2D and 3-D cases. The advantages of the new technique are illustrated by the solution of the 1-D and 2-D impact problems. The new approaches with reduced dispersion can be easily implemented into existing finite element codes and lead to significant reduction in computation time at the same accuracy compared with the standard finite element formulations. Finally, we compare the accuracy of the linear elements with reduced dispersion, the spectral low-and high-order elements as well as the isogeometric elements by the solution of the 1-D impact problem. For all these solutions we use two-stage time integration technique with the filtering stage that removes spurious oscillations and allows an accurate comparison of different space discretization techniques used for elastodynamics. It is also interesting to mention that the amount of numerical dissipation at the filtering stage can be used as a quantitative measure for the comparison of accuracy of the different numerical formulations used for elastodynamics. 759

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“…(6) below) or by the modified integration rule technique (see Eqs. (7), (9) and (11) below). Inserting Eq.…”

Section: The Finite Element Techniques With Reduced Dispersion For Exmentioning

confidence: 97%

“…Due to the space discretization, the exact solution to Eq. (1) contains the numerical dispersion error; e.g., see [7,9,11,26,27,28,30,31,32,14,12] and others. The space discretization error can be decreased by the use of mesh refinement.…”

Section: Introductionmentioning

confidence: 99%

Finite Element Modeling of Linear Elastodynamics Problems With Explicit Time-Integration Methods and Linear Elements With Reduced Dispersion. Comparative Study of Different Finite Element Techniques Used for Elastodynamics.

Idesman¹

2014

Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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“…It computes the internal force vector with the Galerkin formulation of hp-FEA at (p + 1) d Gauss points, but uses the consistent diagonal mass matrix of hp-collocation computed at (p + 1) d GaussLobatto points. This allows for a fully explicit acceleration update as shown in Table 4 and has been widely used in the context of spectral element methods [1,47,49,110]. We multiply the flops per point evaluation given in Appendix A.3 with the total number of quadrature points to obtain the total cost in flops for the formation and update of the global residual vector.…”

Section: Elastodynamics: Cost For An Explicit Time Stepmentioning

confidence: 99%

“…In addition, by using approximation and test functions based on the GLL basis, the consistent mass matrix of hp-collocation is diagonal. This opens the door for fully explicit time integration schemes, and is a well-known property of GLL basis functions in conjunction with Gauss-Lobatto quadrature [1,20,32,[47][48][49].…”

Section: Introductionmentioning

confidence: 99%

A collocated C⁰ finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

Schillinger

Evans

Frischmann

et al. 2014

Numerical Meth Engineering

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We demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, that is, weighted residual formulation, nodal basis functions on Gauss-Lobatto quadrature points, and symmetrization by averaging with the ultra-weak formulation. We quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher-order elastostatic analysis. Furthermore, we illustrate the potential of collocation for efficient higher-order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established. AbstractWe demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, i.e. weighted residual formulation, nodal basis functions on Gauss-Lobatto quadrature points, and symmetrization by averaging with the ultra-weak formulation. We quantify potential gains by comparing the computational efficiency of collocation and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher-order elastostatic analysis. Furthermore we illustrate the potential of collocation for efficient higher-order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established.

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“…[16]), and greater overall simplicity offered by a nodal basis. Further, it has been suggested [6] that hierarchical p-type FEs can have difficulty passing the patch test.…”

Section: Introductionmentioning

confidence: 99%

Legendre spectral finite elements for structural dynamics analysis

Sprague

Geers

2007

Commun. Numer. Meth. Engng.

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SUMMARYSpectral, hierarchical, and h-type finite elements (FEs) are compared in the context of their application to structural dynamics analysis. The Timoshenko beam, which is the 1-D analog of Mindlin-type shell elements, is used for comparison in eigenvalue and transient-response analyses. A detailed formulation of each method is presented to illustrate clearly their fundamental differences. The principal advantages of spectral FEs over low-order h-type elements are (a) far superior accuracy for a fixed number of model degrees of freedom (DOF) and (b) much higher computational efficiency at a fixed accuracy level. The principal advantages over hierarchical p-type elements are (a) a mass matrix that is inherently diagonal as opposed to full, (b) DOF that pertain directly to nodal displacements and rotations, and (c) more efficient tensor-product factorization.

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The solution of elastostatic and elastodynamic problems with Chebyshev spectral finite elements

Cited by 58 publications

References 23 publications

Finite Element Modeling of Linear Elastodynamics Problems With Explicit Time-Integration Methods and Linear Elements With Reduced Dispersion. Comparative Study of Different Finite Element Techniques Used for Elastodynamics.

Finite Element Modeling of Linear Elastodynamics Problems With Explicit Time-Integration Methods and Linear Elements With Reduced Dispersion. Comparative Study of Different Finite Element Techniques Used for Elastodynamics.

A collocated C⁰ finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

Legendre spectral finite elements for structural dynamics analysis

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The solution of elastostatic and elastodynamic problems with Chebyshev spectral finite elements

Cited by 58 publications

References 23 publications

Finite Element Modeling of Linear Elastodynamics Problems With Explicit Time-Integration Methods and Linear Elements With Reduced Dispersion. Comparative Study of Different Finite Element Techniques Used for Elastodynamics.

Finite Element Modeling of Linear Elastodynamics Problems With Explicit Time-Integration Methods and Linear Elements With Reduced Dispersion. Comparative Study of Different Finite Element Techniques Used for Elastodynamics.

A collocated C0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

Legendre spectral finite elements for structural dynamics analysis

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A collocated C⁰ finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics