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Plate Bending by Approximate and Numerical Methods
Published in Eduard Ventsel, Theodor Krauthammer, Thin Plates and Shells, 2001
Eduard Ventsel, Theodor Krauthammer
However, this method has some disadvantages: It can be successfully applied primarily to linear problems.The method requires that a fundamental solution of a governing differential equation or Green’s function be represented in the explicit analytical form. Note, that for the plate bending problems discussed, the fundamental solution is of the very simple analytical form. If the above-mentioned fundamental solution is more awkward than for plate bending problems, the BEM formulation and numerical approximation becomes less efficient.The matrix of the approximating system of linear algebraic equations is complete, unlike the FEM, which causes some difficulties in its numerical implementation.
The boundary element method
Published in Ken P. Chong, Arthur P. Boresi, Sunil Saigal, James D. Lee, Numerical Methods in Mechanics of Materials, 2017
Ken P. Chong, Arthur P. Boresi, Sunil Saigal, James D. Lee
The fundamental solution, also called the Kelvin solution (Love, 2013; Sokolnikoff, 1956), is now introduced. This solution is employed in the boundary element formulation to extract the solution of displacement components from the integral equations of the elastic body. The fundamental solution refers to the solution of the response of a linear, elastic solid of infinite extent due to the application of a point load. Consider a domain of infinite extent shown in Figure 6.4. A unit load is applied at the source point P, and the response at the field point Q is sought. Denoting all quantities related to the fundamental solution with an asterisk (*), the body force
Application of numerical modelling to the analysis of excavations in jointed rock
Published in Hans Peter Rossmanith, Mechanics of Jointed and Faulted Rock, 2020
For the modelling of the far field rock i.e. that part of the rock mass which is far away from the influence of the excavation and for which elastic and homogeneous behaviour can be assumed the best numerical method is the Boundary Element Method. This is because it can deal easily with the fact that this domain is unbounded on at least 5 sides (in the threedimensional case). The method uses a fundamental solution of the governing differential equation to construct particular solutions which satisfy given boundary conditions. It is therefor only necessary to consider the surfaces where such boundary conditions are prescribed. It is not necessary to divide the domain itself into subdomains as would be the case with the Finite Element Method. For the method to work a fundamental solution has to be known. Such solutions are only available for elastic domains which do not contain joints and therefore this model can only be applied to that part of the rock mass where the influence of material nonlinear behaviour and joints is not important. This means that to consider the effect of the infinite rock mass surrounding the near field rock in Figure 3 only the interface between far field and near field rock has to be divided into elements. The boundary conditions applied to the infinite B.E. region are then the ones imposed by the Finite Element mesh. In some cases it may be appropriate to model the entire rock mass as elastic and isotropic. In this case only the surfaces of excavations need to he divided into Boundary Elements. Details of the implementation of the direct Boundary Element method and techniques for coupling both methods have been published previously by Beer (1983, 1986) and need not be repeated here.
Dirichlet-type problems for n-Poisson equation in Clifford analysis
Published in Applicable Analysis, 2022
By direct calculation, it can be observed that is a fundamental solution to the Laplace operator satisfying the following properties: , , for , , for , and ,where is the unit sphere.
Stable determination of an inhomogeneous inclusion in a layered medium
Published in Applicable Analysis, 2022
In this note our purpose is to generalize [7] considering variable coefficients conductivities. The argument to obtain stability deals basically with two main issues: quantitative estimates of unique continuation and singular solutions. The first topic can be derived directly from the constant coefficient case [7] where the authors make use of a three region type inequality developed in [8] based on an ad hoc Carleman estimate [9]. We refer also the interested reader to [10] and [11] for recent developments in this direction. The main difficulty is related with the use of singular solutions. In particular it is crucial establishing the asymptotic behavior. More precisely the key part is to compare the fundamental solution of our operator with the fundamental solution of the Laplace operator (see also [12–14].
The heat kernel of sub-Laplace operator on nilpotent Lie groups of step two
Published in Applicable Analysis, 2021
Der-Chen Chang, Qianqian Kang, Wei Wang
Now we consider a tempered distribution , whose partial Fourier transformation is Similar to [7], we know is locally integrable. Moreover, define It is a tempered distribution because its partial Fourier transformation is locally integrable. Define We claim that is a tempered distribution and prove that is the fundamental solution to the sub-Laplace operator . Before we prove Theorem 1.2, we need a result in [7].