Elastodynamics by BEM: a new direct formulation

Frangi, Attilio

doi:10.1002/(sici)1097-0207(19990630)45:6<721::aid-nme600>3.0.co;2-a

Cited by 22 publications

(2 citation statements)

References 29 publications

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“…We first consider the classical benchmark (see for instance ) of a bar of height equal to 10 and square cross section with side equal to 2. On the lower surface, the Dirichlet boundary data

ū MathClass-rel= 0

is enforced, whereas the upper surface is subjected to a uniform Neumann condition

\overset{p}{true} MathClass-rel= H [t]

.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Instead, it is expedient to consider, for k = 0, ⋯ , N Δ t − 1, the following hat‐shaped time shape function

{\overset{ψ}{true}}_{k} (t) MathClass-rel= R (t MathClass-bin- t_{k}) MathClass-bin- 2 R (t MathClass-bin- t_{k MathClass-bin+ 1}) MathClass-bin+ R (t MathClass-bin- t_{k MathClass-bin+ 2}) MathClass-punc,

which preserves the cut‐off property of all BEM matrix blocks related to the unknown u 1 ; hence, in particular, as proved in , of the elements coming from the discretization of

MathClass-rel< D u_{1} MathClass-punc, {\overset{v}{true}}_{1} {MathClass-rel>}_{Γ_{1 MathClass-punc, N}}

, which, after a regularizing procedure of the hypersingular bilinear form and an analytical integration in time, are of the form

MathClass-bin- \frac{1}{4 π {(Δt)}^{2}} {MathClass-op\sum}_{α MathClass-punc, β MathClass-punc, λ MathClass-rel= 0}^{1} {(MathClass-bin- 1)}^{α MathClass-bin+ β MathClass-bin+ λ} {MathClass-op\int}_{Γ_{1 MathClass-punc, N}} {MathClass-op\int}_{Γ_{1 MathClass-punc, N}} \frac{1}{r} H [Δ_{h MathClass-bin+ α MathClass-punc, k MathClass-bin+ β MathClass-bin+ λ} MathClass-bin- \frac{r}{c_{1}}] scriptD (r MathClass-punc, Δ_{h MathClass-bin+ α MathClass-punc, k MathClass-bin+ β MathClass-bin+ λ}

…”

Section: Extensions Of the Model Problemmentioning

confidence: 99%

See 1 more Smart Citation

Energetic BEM–FEM coupling for wave propagation in 3D multidomains

Aimi

Diligenti

Frangi

et al. 2013

Numerical Meth Engineering

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SUMMARYStarting from a recently developed energetic space–time weak formulation of boundary integral equations for wave propagation problems, a coupling algorithm is presented, which allows a flexible use of FEMs and BEMs as local discretization techniques. Emphasis is given to theoretical and experimental analysis of the stability of the proposed method. Several numerical results on model problems are presented and discussed, showing that both bounded and unbounded three‐dimensional domains can be efficiently addressed. Copyright © 2013 John Wiley & Sons, Ltd.

show abstract

“…We first consider the classical benchmark (see for instance ) of a bar of height equal to 10 and square cross section with side equal to 2. On the lower surface, the Dirichlet boundary data

ū MathClass-rel= 0

is enforced, whereas the upper surface is subjected to a uniform Neumann condition

\overset{p}{true} MathClass-rel= H [t]

.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Instead, it is expedient to consider, for k = 0, ⋯ , N Δ t − 1, the following hat‐shaped time shape function

{\overset{ψ}{true}}_{k} (t) MathClass-rel= R (t MathClass-bin- t_{k}) MathClass-bin- 2 R (t MathClass-bin- t_{k MathClass-bin+ 1}) MathClass-bin+ R (t MathClass-bin- t_{k MathClass-bin+ 2}) MathClass-punc,

which preserves the cut‐off property of all BEM matrix blocks related to the unknown u 1 ; hence, in particular, as proved in , of the elements coming from the discretization of