# A comparative study on three boundary element method approaches to problems in elastodynamics

Abstract: In this study, three boundary element method approaches are compared by solving a typical problem in linear elastodynamics. The first approach formulates and solves the problem in the real time domain in conjunction with a time-stepping algorithm. The other two approaches formulate and solve the problem in the Laplace and the Fourier transformed domains, which necessitates a numerical inversion of the results to the time domain. The results obtained compare favourably with available analytic solutions. A detailed tabulation of the computer time and memory requirements of each approach is presented.

## Summary (3 min read)

### INTRODUCTION

- An approach in which the integral equation formulation is applied to the Laplace transform of the governing equations of motion for the two-dimensional case.
- All of the aforementioned cases share some common traits.
- Following that, the results drawn from applying the three aforementioned BEM approaches to the same problem, which is shown in Figure 1 , are discussed in detail.

### STATEMENT OF THE PROBLEM AND DEFINITIONS

- Let V denote the area occupied by a given body and S denote its bounding surface.
- Associated with each surface point is an outward pointing unit normal vector n.
- Under the assumption of small displacement theory and linear elastic isotropic and homo~eneous material behaviour, the equations of motion of the body are (1) where u;(x, t) is the displacement vector and[; is the body force vector per unit mass.
- The propagation velocities of the pressure and shear waves in the body are given, respectively, as ci=(A+2M)/p; c~=IJ.IP (2) where A and IJ. are the Lame constants and p is the mass density.
- Furthermore, commas indicate space differentiation, dots indicate time differentiation and repeated indices imply the summation convention.

### The Laplace transform

- A Laplace transform of the equations of motion (1), which are hyperbolic partial differential equations, results in the following system of elliptic partial differential equations: ( EQUATION which is more amenable to numerical solutions.
- It should be noted in passing that the Navier equations of equilibrium are also of elliptic nature, i.e. they are similar in form to equation (7).
- This series of solutions must finally be numerically inverted back to the original time domain.
- Also, the body forces are taken equal to zero as well.
- These assumptions are made for the sake of convenience and no additional conceptual difficulty in the solution procedure is encountered if these assumptions are relaxed.

### INTEGRAL EQUATION FORMULATION

- It is well known (DeHoop/ 1 Wheeler and Sternberg 22 and Kupradze 4 ) that a system of partial differential equations along with the appropriate boundary and initial conditions may be cast in an integral equation form.
- This holds true for either the system of equations (1)- (5) or the system of equations ( 7)-( 9), depending on whether one wishes to establish an integral equation formulation in either the time domain or in a transformed domain.
- The basis for an integral equation formulation in elastodynamics is the dynamic extension of Betti's reciprocal theorem.
- The obvious choice for one of the two elastodynamic states is the unknown state the authors are seeking to find.
- For more details, an appropriate reference is Banerjee and Butterfield.

### Time domain integral equation formulations

- In (11), the prime symbol used as a superscript denotes three-dimensional quantities with the pertinent subscripts ranging from 1 to 3.
- Finally, F ;i represents the resulting tractions due to the same unit impulse and can be obtained from G ~i through the use of the constitutive equations and surface normals.
- Along with details of the solution procedure.
- The approach delineated there is essentially a three-dimensional formulation applied to an infinitely long cylindrical body whose longitudinal axis coincides with the Z axis of the cartesian co-ordinate system .
- The problem, of course, remains a plane strain one.

### Laplace transform domain integral e'quation formulation

- In the Laplace transform domain, the Green's function can be found in Cruse and Rizzo 8 and Nowacki 24 and the singular integral equations are of the form EQUATION.
- The explanatory notes following equation ( 11) apply here as well.
- It should be reminded at this point that bars over a quantity denote the Laplace transform of that quantity.
- The solution procedure in the Laplace transform domain is elaborated in detail in Manolis and Beskos 10 and only a few remarks will be made here.

### Fourier transform domain integral equation formulation

- For this case, the fast Fourier transform algorithm of Cooley and Tukey, 26 which has been extensively used by numerous investigators in the past is employed.
- Since the upper and lower limits in (18) are replaced by 0 and !1 (the maximum frequency of interest), the well-known problem associated with the finite Fourier transform, namely the approximation a non-periodic motion by a periodic one, arises.
- In the particular example used here, this numerical difficulty was not encountered.
- Some details on this subject can be found in Kobayashi and Nishimura.

### NUMERICAL EXAMPLE

- This section describes the detailed solution of a numerical example that serves as the vehicle for a comparison study of the three BEM approaches.
- It can be observed from Figures 4 and 6 that the results obtained from the three BEM formulations agree reasonably well with the results of analytic-numerical approaches.
- The true behaviour at this location is probably better represented by the results obtained by Baron and Matthews, because when the P-wave pulse first reaches the cavity, the points around(}= oo act as a rigid wall and as a consequence the applied stress iiy should double.
- Also, it should be mentioned that all computations were performed on a CDC Cyber 176 computing machine.

### CONCLUSIONS

- On the basis of the preceding presentation, the following conclusions can be drawn: 1. Three general BEM formulations for solving problems in transient elastodynamics involving bodies of arbitrary shape under conditions of plane stress or plane strain are presented.
- All three formulations are viable alternatives for the solution of general elastodynamic problems.
- Integral transform formulations tend to reach a plateau corresponding to the solution of the static equivalent of the problem, while time domain formulations tend to diverge after a very large number of time steps.
- To circumvent this economic difficulty, one may employ, in the beginning stages of an analysis/design procedure, simplified methods of analysis and resort to the powerful BEM approaches at the end of the procedure.
- Some further advantages of the time domain BEM are that it is more suitable for extension to nonlinear material behaviour through the use of incremental and iteration schemes and that the extension to three-dimensional problems is straightforward.

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