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A shooting approach to the scaled boundary finite element equations of elastodynamics in the frequency domain

Daneshyar, A ; Sharif University of Technology | 2021

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  1. Type of Document: Article
  2. DOI: 10.1016/j.cma.2021.114170
  3. Publisher: Elsevier B.V , 2021
  4. Abstract:
  5. Dealing with numerical analysis of problems, especially ones with semi-infinite boundaries, scaled boundary finite element method has emerged as one of the efficient tools for the task. Combining the exactness of strong forms with the flexibility of weak formulations makes the method an improvement to its predecessors. Problem with the method arises when the analytical solution of the semi-discretized system is not available, which is the case for numerous problems. In the most recent attempt to solve the issue, a shooting method was proposed for elastostatic problems. Generality of the method removes any concerns regarding the type of governing equations since it no longer needs any analytical solutions. In this study, the method is further developed to deal with elastodynamic problems of viscoelastic media in the frequency domain, for which the closed-form solution is not available. Adopting the shooting method, the two-point boundary value problem of each subdomain is treated as a second-order initial value problem. Boundary conditions of the bounded subdomains are formulated by enforcing equilibrium at both ends. On the other hand, in contrast to linear elasticity, neither forces nor displacements vanish at infinity. As a result, a special treatment is required to deal with the boundary conditions of the unbounded subdomains. A Runge–Kutta–Nyström method, which consists of an embedded pair of adjacent order, is utilized to integrate the strong form of oscillatory equations enabling local error estimation and adaptive step-size control. This guarantees efficiency since the integration proceeds not by equal step-sizes, but by equal user-defined local errors. In the first step, analytical solution of a viscoelastic half-space subjected to a harmonic strip load is used for the verification process. First, the half-space is discretized by Voronoi cells and each truncated edge is defined using an unbounded subdomain. Next, one bounded subdomain is used to define the whole bounded area, and one unbounded subdomain resembles the radiation condition of the far-field boundaries. For the cases involving only bounded subdomains, three other numerical tests, including a perforated plate, an L-shaped panel, and a bi-material specimen are employed, and the effect of discretization method on the results is analyzed. The method is a huge step forward since the size of subdomains are not restricted, far-field boundaries can be arbitrary-shaped, hysteretic damping can be naturally included, no additional unknowns are introduced, no compound matrices are defined, no preconditioning matrices are needed, and a simple unified approach is used for both bounded and unbounded subdomains. © 2021 Elsevier B.V
  6. Keywords:
  7. Boundary conditions ; Discrete event simulation ; Errors ; Finite element method ; Geometry ; Initial value problems ; Numerical methods ; Runge Kutta methods ; Viscoelasticity ; Elasto-dynamics ; Far-field ; Field boundaries ; Frequency domains ; Runge-kutta-Nystrom methods ; Scaled boundary finite elements methods ; Shooting methods ; Shooting technique ; Strong form ; Subdomain ; Frequency domain analysis
  8. Source: Computer Methods in Applied Mechanics and Engineering ; Volume 387 , 2021 ; 00457825 (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/abs/pii/S0045782521005016