Solution of Dynamic Problems of Acoustics by the Finite-Difference Method in the Time Domain: Basic Relationships and Analysis of Errors

Abstract

The possibility of applying the finite-difference time-domain method for calculating wave processes in a solid is considered. The reliability of this method in acoustical problems is tested. As a result of testing the calculation schemes, the optimum algorithms are found and the conditions are determined under which highly accurate calculation results are attained.The line of research based on numerical simulations of complex dynamic problems has been actively developed since the advent of high-performance computers. Partial differential equations are solved using the finite-difference time-domain method (FDTD method). The FDTD method was proposed approximately 30 years ago, but it has been widely used only in the last decade.At present the FDTD method is one of the most popular techniques for numerically solving problems of electrodynamics. A great number of papers devoted to various aspects of the formation of electromagnetic fields and propagation of radio waves have been published. Some of these papers [1-5] that attracted the author's interest are presented in the list of references.The interests of researchers in acoustic problems are concentrated mainly around the problem of ultrasonic diagnostics in medicine (see, e.g., [6][7][8][9][10][11][12]). Unfortunately, studies aimed at solving problems of flaw detection are virtually nonexistent.The FDTD method has a number of obvious advantages: (i) simulation of nonstationary wave processes in viscously elastic, inhomogeneous anisotropic, and nonlinear media;(ii) the possibility of calculating waves in problems with arbitrarily complex boundary conditions; (iii) obtainment of time-dependent solutions (with certain discrete step ∆ t ); and (iv) a comparatively simple calculation algorithm that can easily be performed by a computer. The FDTD method has certain drawbacks. It has a tendency to accumulate calculation errors. Due to errors in rounding numbers and in approximations adopted for calculations, noise in solutions and false wave processes arise.The formulation of the FDTD method is based on the sampling of differential equations in space and time. Solutions are sought in nodes of a multidimensional mesh. A problem of applying the FDTD method is the necessity for use of powerful high-performance computers with a RAM capacity of hundreds or thousands of megabytes. Therefore, it becomes necessary to develop algorithms that allow calculations on meshes with large widths. This allows a decrease in the capacity of the required computer RAM and simultaneously shortens the calculation time.Therefore, before solving important problems of ultrasonic flaw detection, it is necessary to verify the reliability of the FDTD method in acoustic problems, find optimum algorithms, and determine the conditions under which the results are calculated with a high accuracy.The calculation schemes are formulated and errors of the algorithms are analyzed using the solutions to a one-dimensional equation in a medium with internal friction (1) where C is the velocity of sound and η is...