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Integral and Integro-Differential Equations
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
One of the most popular integral transforms is the Laplace transform, which is applicable to problems in the semi-infinite domain. It can be shown that the Fourier transform to be presented next is equivalent to the Laplace transform for the infinite domain (Sneddon, 1951). The Laplace transform was originally proposed by Euler and developed to the present form by Laplace. It had been known as operational calculus and its application in engineering is mainly publicized by Oliver Heaviside in the 1980s. The kernel of the Laplace transform is K(s,t)=e-st $$ K(s,t) = e^{{ - st}} $$
Implementation, Modeling, and Related Issues
Published in Victor N. Kaliakin, Introduction to Approximate Solution Techniques, Numerical Modeling, and Finite Element Methods, 2018
This brings to light the important observation that, in many physical applications, elements with special properties are required to achieve maximum accuracy. For example, certain singular or localization elements for modeling point and line singularities have been developed [221, 7, 57, 541, 508, 251, 422, 67]. A second example are so-called infinite elements [72, 73, 61, 595, 128, 374, 148] that allow for accurate representation of semi-infinite domains without resorting to boundary solution procedures. The final example are interface elements that facilitate the modeling of relative displacements, contact, stresses, etc. along material interfaces [202, 232, 196, 282, 140, 63, 193, 485, 276],
Examples of Soil-Tool Interaction
Published in Jie Shen, Radhey Lal Kushwaha, Soil-Machine Interactions, 2017
Theoretically, a soil-tool system should be defined on a semi-infinite domain. Conventional finite element method idealizes this semi-infinite domain to an enough large domain and then gives the corresponding boundary conditions. It is assumed that the influence from outside the specified domain is negligible. But this is true only for an enough large domain. For three dimensional analysis, in some cases computation time cost may become intolerable. Therefore, an effective way to dramatically reduce the total number of d.o.f. is of importance.
Site Amplification Response by a Dual Reciprocity Boundary Element Method
Published in Journal of Earthquake Engineering, 2022
Bahman Ansari, Alireza Firoozfar
Boundary element analysis is another important numerical method especially when dealing with semi-infinite problems. Site effect analysis with boundary element method is simple because the meshing procedure only includes the boundaries of the model and much lesser computational effort is required for solving the equations. Different researchers used boundary elements to analyze the effects of the topography on the seismic site response. Pinelopi, Kontoni, and Beskos (1986) used a uniform half-plane boundary element method for obtaining ground surface responses due to existence of the embedded cavities. With using the boundary element analysis, Alvarez-Rubio et al. (2005) studied the effects of the complex topography and soil layering on the ground surface seismic responses. Alielahi, Kamalian, and Adampira (2015) computed the ground surface amplification responses due to the propagation of in-plane P and SV waves, in the presence of the subsurface structures such as cavities. Amplification due to the existence of the topographic features especially in a porous medium has been studied by Liu et al. (2017). Panji and Ansari (2017a, 2017b, 2019) developed and implemented a time domain half-plane boundary element solution for obtaining ground surface responses due to the propagation of the anti-plane SH wave. Ba and Yin (2016) proposed a multi-domain indirect boundary element method (IBEM) to study the wave scattering of plane SH waves by complex local site in a layered half-space. Their method uses both full-space and layered half-space Green’s functions as fundamental solutions, which present a coupled method of full-space and half-space IBEM. In another research, Ba et al. (2020) extended the coupled IBEM method for investigating the wave scattering of plane P, SV, and SH waves by a 3D alluvial basin embedded in a multilayered half-space.