Boundary integral Equations Method in two- and three-dimensional problems of elastodynamics

Alekseyeva, L.A.; Zakir’yanova, G. K.; Dildabayev, Sh. A.; Zhanbyrbayev, A. B.

doi:10.1007/bf00350533

Cited by 12 publications

(5 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(1) in a generalized sense. The natural requirement of the continuity of the solutions at transition through wave front F u i x, t ðÞ ½ F ¼ 0 (8) vanishes only two last composed right parts of Eq. (7).…”

Section: R Nþ1mentioning

confidence: 99%

“…These equations can be numerically solved using the boundary element method. In special cases of nonstationary boundary value problems in elasticity theory (M ¼ N ¼ 2, 3), these equations were solved in [4,[6][7][8].…”

Section: Theorem 53 If a Classical Solution Of The Dirichlet (Neumamentioning

confidence: 99%

“…For systems of hyperbolic equations, the BIE method is developed. Here, the ideas for solving nonstationary BVPs for the wave equations in multidimensional space [3,4] are used and the methods were elaborated for boundary value problems of dynamics of elastic bodies [5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations of Mathematical Physics

Alexeyeva¹,

Zakir’yanova²

2020

Mathematical Theorems - Boundary Value Problems and Approximations

View full text Add to dashboard Cite

The method of boundary integral equations is developed for solving the nonstationary boundary value problems (BVP) for strictly hyperbolic systems of second-order equations, which are characteristic for description of anisotropic media dynamics. The generalized functions method is used for the construction of their solutions in spaces of generalized vector functions of different dimensions. The Green tensors of these systems and new fundamental tensors, based on it, are obtained to construct the dynamic analogues of Gauss, Kirchhoff, and Green formulas. The generalized solution of BVP has been constructed, including shock waves. Using the properties of integrals kernels, the singular boundary integral equations are constructed which resolve BVP. The uniqueness of BVP solution has been proved.

show abstract

Section: R Nþ1mentioning

confidence: 99%

Section: Theorem 53 If a Classical Solution Of The Dirichlet (Neumamentioning

confidence: 99%

See 1 more Smart Citation

Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations of Mathematical Physics

Alexeyeva¹,

Zakir’yanova²

2020

Mathematical Theorems - Boundary Value Problems and Approximations

View full text Add to dashboard Cite

show abstract

“…We have used an isoparametric curvilinear approach, as for the EM case, but the stability issues are not really dependent on this. Again, the lengthy details are available elsewhere [34][35][36]. In (7) we see the displacement u i (a three vector) at any node i depends on (a) the displacement at the present k þ 1 timestep at 'all' other nodes j (although most matrices T will have all zero elements, except for nodes j close to node i) and (b) the displacement at all other nodes at all earlier timesteps k þ 1 À w for w ¼ 1 to W ; the total number of timesteps in a transit of the body at the (lower) wave speed.…”

Section: Elastodynamicsmentioning

confidence: 99%

The stability of integral equation time‐domain scattering computations for three‐dimensional scattering; similarities and differences between electrodynamic and elastodynamic computations

Walker

Bluck

Chatzis

2002

Int J Numerical Modelling

View full text Add to dashboard Cite

SUMMARYTime-domain integral equation analyses are prone to instabilities, in a range of applications areas including acoustics, electrodynamics and elastodynamics, and a variety of retrospective averaging schemes have been proposed to improve matters. In this paper, we investigate stability behaviour, in parallel, in electrodynamic and elastodynamic cases. It is observed empirically that the tendency to instability is increased as the treatment becomes more nearly explicit. The timestepping procedure is recast in a recursive matrix formulation, and it is shown that it is the eigenvalues of this large matrix which determine the longterm stability behaviour. This treatment is then extended to cover general averaging schemes, allowing the likely effectiveness of such schemes to be assessed. Simple modelling rules, which for all practical purposes will ensure stability, are presented.

show abstract

“…In Refs. [2][3][4][5][6], the MGF was developed to solve nonstationary and stationary boundary value problems of the theory of elasticity, thermoelasticity, and electrodynamics and initial-boundary value problems for hyperbolic equations systems which are typical to the mathematical physics [7].…”

Section: Introductionmentioning

confidence: 99%

Boundary Integral Equations of no Stationary Boundary Value Problems for the Klein-Gordon Equation

Aigulim¹,

Assiyat²

2020

Mathematical Theorems - Boundary Value Problems and Approximations

View full text Add to dashboard Cite

The non-stationary boundary value problems for Klein-Gordon equation with Dirichlet or Neumann conditions on the boundary of the domain of definition are considered; a uniqueness of boundary value problems is proved. Based on the generalized functions method, boundary integral equations method is developed to solve the posed problems in strengths of shock waves. Dynamic analogs of Green's formulas for solutions in the space of generalized functions are obtained and their regular integral representations are constructed in 2D and 3D over space cases. The singular boundary integral equations are obtained which resolve these tasks.

show abstract

Boundary integral Equations Method in two- and three-dimensional problems of elastodynamics

Cited by 12 publications

References 6 publications

Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations of Mathematical Physics

Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations of Mathematical Physics

The stability of integral equation time‐domain scattering computations for three‐dimensional scattering; similarities and differences between electrodynamic and elastodynamic computations

Boundary Integral Equations of no Stationary Boundary Value Problems for the Klein-Gordon Equation

Contact Info

Product

Resources

About