On mass lumping and explicit dynamics in the scaled boundary finite element method

Gravenkamp, Hauke; Song, Chongmin; Zhang, Junqi

doi:10.1016/j.cma.2020.113274

Cited by 33 publications

(12 citation statements)

References 49 publications

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“…The total mass of the system is conserved in the formulation of the SBFEM. A detailed explanation and verification of this mass lumping approach has been reported by Gravenkamp et al [47].…”

Section: Mass Lumping In the Sbfemsupporting

confidence: 53%

“…More detailed investigations regarding the use of LMMs in the framework of the SBFEM are discussed in Ref. [47]. A convergence study of all element types is performed, and the L 2 norm of the relative error e in the first 100 frequencies is presented in Fig.…”

Section: Eigenfrequencies Of a Cubementioning

confidence: 99%

“…This problem is equivalent to a 1D wave propagation problem; therefore, Duhamel's integral is applied to obtain a reference solution for the displacement response, the details of which can be found in Ref. [47].…”

Section: Wave Propagation In a Beammentioning

confidence: 99%

“…An advantage of the octree meshes benefiting parallel computing is that only a limited number of octree patterns are present, and therefore, the element stiffness and mass matrices can be pre-computed. The nodal force vector is calculated element-wise without the need for assembling the global stiffness matrix, while the global mass matrix is lumped into a well-conditioned diagonal matrix based on a recently developed technique [47]. All elements of the same shape (i.e., same nodal pattern) are grouped, and their nodal displacement vectors are assembled as a matrix.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A massively parallel explicit solver for elasto-dynamic problems exploiting octree meshes

Zhang,

Ankit,

Gravenkamp

et al. 2021

Preprint

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Typical areas of application of explicit dynamics are impact, crash test, and most importantly, wave propagation simulations. Due to the numerically highly demanding nature of these problems, efficient automatic mesh generators and transient solvers are required. To this end, a parallel explicit solver exploiting the advantages of balanced octree meshes is introduced. To avoid the hanging nodes problem encountered in standard finite element analysis (FEA), the scaled boundary finite element method (SBFEM) is deployed as a spatial discretization scheme. Consequently, arbitrarily shaped star-convex polyhedral elements are straightforwardly generated. Considering the scaling and transformation of octree cells, the stiffness and mass matrices of a limited number of unique cell patterns are pre-computed. A recently proposed mass lumping technique is extended to 3D yielding a well-conditioned diagonal mass matrix. This enables us to leverage the advantages of explicit time integrator, i.e., it is possible to efficiently compute the nodal displacements without the need for solving a system of linear equations. We implement the proposed scheme together with a central difference method (CDM) in a distributed computing environment. The performance of our parallel explicit solver is evaluated by means of several numerical benchmark examples, including complex geometries and various practical applications. A significant speedup is observed for these examples with up to one billion of degrees of freedom and running on up to 16,384 computing cores.

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Section: Mass Lumping In the Sbfemsupporting

confidence: 53%

Section: Eigenfrequencies Of a Cubementioning

confidence: 99%

Section: Wave Propagation In a Beammentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A massively parallel explicit solver for elasto-dynamic problems exploiting octree meshes

Zhang,

Ankit,

Gravenkamp

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…This was successfully applied to simulate Lamb waves on 2 and 3D aluminium plates with holes [40] and piezo-electric sensors for SHM [41]. Among other high order methods suitable for dynamic analysis [42,43,44,45,46], the SEM is often one of the preferred approaches [47,48,49], since use of Gauss-Lobatto-Legendre (GLL) integration points delivers a variationally consistent diagonal mass matrix, without incurring loss of accuracy [48,50,51] or the need for additional lumping procedures. However, when decoupled geometrical descriptions as in the SCM or the XFEM are employed, special integration rules are applied for elements intersected by a boundary, thus eliminating the diagonal property of the mass matrix.…”

Section: Introductionmentioning

confidence: 99%

Moment fitted cut spectral elements for explicit analysis of guided wave propagation

Nicoli,

Agathos,

Chatzi

2021

Preprint

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In this work, a method for the simulation of guided wave propagation in solids defined by implicit surfaces is presented. The method employs structured grids of spectral elements in combination to a fictitious domain approach to represent complex geometrical features through singed distance functions. A novel approach, based on moment fitting, is introduced to restore the diagonal mass matrix property in elements intersected by interfaces, thus enabling the use of explicit time integrators. Since this approach can lead to significantly decreased critical time steps for intersected elements, a "leap-frog" algorithm is employed to locally comply with this condition, thus introducing only a small computational overhead. The resulting method is tested through a series of numerical examples of increasing complexity, where it is shown that it offers increased accuracy compared to other similar approaches. Due to these improvements, components of interest for SHM-related tasks can be effectively discretized, while maintaining a performance comparable or only slightly worse than the standard spectral element method.

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The scaled boundary finite element method for dispersive wave propagation in higher‐order continua

Daneshyar

Sotoudeh

Ghaemian

2022

Numerical Meth Engineering

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The classical theory of elasticity relies on the perfect structure of matter. No matter how small a medium is, it always stays homogeneous-but the reality is different. Even though the classical continuum mechanics suffices well for describing phenomena that evolve at super-microscopic scales, it ceases to reproduce reasonable responses when the size effects are pronounced. It is due to the local constitutive equations of classical continuum mechanics, which are devoid of any length scales associated with the underlying microstructure.The gradient-dependent theory of elasticity redresses this shortcoming by incorporating the kinematic quantities of the corresponding representative volume element by enriching the differential equations with higher-order spatial and temporal derivatives. In this study, the scaled boundary finite element method is formulated for the equations of motion with higher-order inertia terms. To this end, the semi-discretized scaled boundary finite element equations of the medium are derived by introducing the scaled boundary transformation of geometry to the gradient-enriched equations of motion and applying the Galerkin method of weighted residuals. It is shown that the well-established available solution methods are incapable of handling the frequency-domain representation of the derived formulation. Accordingly, a numerical solution method based on the shooting technique is proposed. The solution procedure is formulated for general numerical integration schemes via the infinite Taylor series. The evolution of the impedance-diffusion matrix and contribution of inter-subdomain forces and tractions are extracted. Four different numerical integration methods are employed to describe the solution procedure. In addition, a comparison regarding their computational efficiency is presented. For the sake of verification, the scaled boundary finite element solutions of three numerical examples are compared with reference solutions that are obtained using finite element models with extremely fine meshes. The convergence trends are also presented for both h-and p-refinement methods. The results demonstrate the capability of the proposed formulation in reproducing accurate responses.

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On mass lumping and explicit dynamics in the scaled boundary finite element method

Cited by 33 publications

References 49 publications

A massively parallel explicit solver for elasto-dynamic problems exploiting octree meshes

A massively parallel explicit solver for elasto-dynamic problems exploiting octree meshes

Moment fitted cut spectral elements for explicit analysis of guided wave propagation

The scaled boundary finite element method for dispersive wave propagation in higher‐order continua

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