FFT-Based Methods for Computational Contact Mechanics

Abstract

Computational contact mechanics seeks for numerical solutions to contact area, pressure, deformation, and stresses, as well as flash temperature, in response to the interaction of two bodies. The materials of the bodies may be homogeneous or inhomogeneous, isotropic or anisotropic, layered or functionally graded, elastic, elastoplastic, or viscoelastic, and the physical interactions may be subjected to a single field or multiple fields. The contact geometry can be cylindrical, point (circular or elliptical), or nominally flat-to-flat. With reasonable simplifications, the mathematical nature of the relationship between a surface excitation and a body response for an elastic contact problem is either in the form of a convolution or correlation, making it possible to formulate and solve the contact problem by means of an efficient Fourier-transform algorithm. Green's function inside such a convolution or correlation form is the fundamental solution to an elementary problem, and if explicitly available, it can be integrated over a region, or an element, to obtain influence coefficients (ICs). Either the problem itself or Green's functions/ICs can be transformed into a space-related frequency domain, via a Fourier transform algorithm, to formulate a frequency-domain solution for contact problems. This approach converts the original tedious integration operation into multiplication accompanied by Fourier and inverse Fourier transforms, and thus a great computational efficiency is achieved. The conversion between ICs and frequency-response functions facilitate the solutions to problems with no explicit space-domain Green's function. This paper summarizes different algorithms involving the fast Fourier transform (FFT), developed for different contact problems, error control, as well as solutions to the problems involving different contact geometries, different types of materials, and different physical issues. The related works suggest that (i) a proper FFT algorithm should be used for each of the cylindrical, point, and nominally flat-flat contact problems, and then (ii) the FFT-based algorithms are accurate and efficient. In most cases, the ICs from the 0order shape function can be applied to achieve satisfactory accuracy and efficiency if (i) is guaranteed.

Simulation of Adhesive Contact of Soft Microfibrils

“…However, the convolution of integral ( 7) or ( 8) can be solved by the fast Fourier transform (FFT), which reduces the complexity greatly from o(N 4 ) to o(N 2 logN 2 ). This technique has been applied by many researchers [26]. Before carrying out the FFT, the matrix of the influence coefficient is usually constructed in the "convolution" form with the same dimension N × N.…”

Frictional contact problem of one-dimensional hexagonal piezoelectric quasicrystals layer

“…The discrete Fourier transforms of ICs are calculated from continuous ICs with the aliasing treatment [35,36]…”

FFT-Assisted Algorithms for 3d Line-Contact Problems

“…1. Compute the influence coefficients and treat the resulting series with zero-padding and wrap-around order, to accomplish the required circular nature [19] of the convolution series. 2.…”

“…This author [18] also applied spectral methods to simulate the contact process of viscoelastic materials in finite length line-contacts. Reviews on the application of the FFT in tribological modelling can be found in [10,19]. In the light of these recent developments in the spectral analysis of contact problems, two modifications of the state-of-the-art Discrete Convolution Fast Fourier Transform (DCFFT) algorithm are advanced in this paper.…”