A Fast Method for Solving Contact Plasticity in a Half Space

2011 ◽  
Author(s):  
Zhanjiang Wang ◽  
Xiaoqing Jin ◽  
Shuangbiao Liu ◽  
Leon M. Keer ◽  
Jian Cao ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Residual Stresses ◽  
Fast Fourier Transform ◽  
Half Space ◽  
Three Dimensional ◽  
New Method ◽  
Fast Method ◽  
Discrete Convolution ◽  
Influence Coefficients ◽  
The One

This paper presents a new method of contact plasticity analysis based on Galerkin vectors to solve the eigenstresses due to eigenstrain. The influence coefficients relating eigenstrains to eigenstresses thus can be divided into four terms the one due to the eigenstrains in the full space, others due to the mirrored eigenstrains in the mirror half space. Each term can be solved fast and efficient by using the three-dimensional discrete convolution and fast Fourier transform (DC-FFT) or the three-dimensional discrete correlation and fast Fourier transform (DCR-FFT). The new method is used to analyze the contact plastic residual stresses in half space.

Elastic Fields due to Eigenstrains in a Half-Space

2005 ◽  
Vol 72 (6) ◽  
pp. 871-878 ◽  
Author(s):  
Shuangbiao Liu ◽  
Qian Wang
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Half Space ◽  
Numerical Algorithms ◽  
Elastic Field ◽  
Contact Problems ◽  
Fast Fourier Transform Algorithm ◽  
Elastic Fields ◽  
Influence Coefficients ◽  
Residual Strains

Engineering components inevitably encounter various eigenstrains, such as thermal expansion strains, residual strains, and plastic strains. In this paper, a set of formulas for the analytical solutions to cases of uniform eigenstrains in a cuboidal region-influence coefficients, is presented in terms of derivatives of four key integrals. The linear elastic field caused by arbitrarily distributed eigenstrains in a half-space is thus evaluated by the discrete correlation and fast Fourier transform algorithm, along with the discrete convolution and fast Fourier transform algorithm. By taking advantage of both the convolution and correlation characteristics of the problem, the formulas of influence coefficients and the numerical algorithms are expected to enable efficient and accurate numerical analyses for problems having nonuniform distribution of eigenstrains and for contact problems.

Studying Contact Stress Fields Caused by Surface Tractions With a Discrete Convolution and Fast Fourier Transform Algorithm

2001 ◽  
Vol 124 (1) ◽  
pp. 36-45 ◽  
Author(s):  
Shuangbiao Liu ◽  
Qian Wang
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Frequency Response ◽  
High Efficiency ◽  
Contact Stresses ◽  
Influence Coefficient ◽  
Response Functions ◽  
Frequency Response Functions ◽  
Discrete Convolution ◽  
Influence Coefficients

The knowledge of contact stresses is critical to the design of a tribological element. It is necessary to keep improving contact models and develop efficient numerical methods for contact studies, particularly for the analysis involving coated bodies with rough surfaces. The fast Fourier Transform technique is likely to play an important role in contact analyses. It has been shown that the accuracy in an algorithm with the fast Fourier Transform is closely related to the convolution theorem employed. The algorithm of the discrete convolution and fast Fourier Transform, named the DC-FFT algorithm includes two routes of problem solving: DC-FFT/Influence coefficients/Green’s function for the cases with known Green’s functions and DC-FFT/Influence coefficient/conversion, if frequency response functions are known. This paper explores the method for the accurate conversion for influence coefficients from frequency response functions, further improves the DC-FFT algorithm, and applies this algorithm to analyze the contact stresses in an elastic body under pressure and shear tractions for high efficiency and accuracy. A set of general formulas of the frequency response function for the elastic field is derived and verified. Application examples are presented and discussed.

Three-Dimensional Semiperiodic Line Contact–Periodic in Contact Length Direction

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
Shuangbiao Liu ◽  
Diann Y. Hua
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Three Dimensional ◽  
Contact Problems ◽  
Contact Length ◽  
Computational Domain ◽  
Line Contact ◽  
Discrete Convolution ◽  
Length Direction ◽  
Contact Domain

Line contact problems, such as those seen in spur gears and cam-roller follower systems, are often simplified with the plane-strain assumption and thus modeled by two-dimensional equations. However, in order to address the effects of roughness and textured surfaces, three-dimensional modeling is necessary. The challenge arises when the contact domain is several orders of magnitude greater than the grid size needed to properly describe the surface roughness or texture. Considering the surface geometry of a so-called “line contact,” the contact domain is nonperiodic in contact width direction, but it can be treated as periodic in the contact length direction–semiperiodic line contact problem. Thus, only a section of the entire contact domain is used as the computational domain with a much-reduced size. Based on an in-depth investigation of available algorithms, DC-FFTS and DC-CC-FFT algorithms are proposed. The DC-FFTS algorithm is a modified discrete convolution and fast Fourier transform algorithm with superposition of influence coefficients. The DC-CC-FFT algorithm is a hybrid fast Fourier transform based algorithm, which combines the discrete convolution–FFT and the continuous convolution–FFT methods. The proposed algorithms are used to solve three-dimensional displacement, contact pressure, and stresses for line contact problems. The results are compared with the other available algorithms from literature. The accuracy and efficiency of different algorithms are discussed.

scholarly journals On Performance of Sparse Fast Fourier Transform Algorithms Using the Aliasing Filter

Electronics ◽  
2021 ◽  
Vol 10 (9) ◽  
pp. 1117
Author(s):  
Bin Li ◽  
Zhikang Jiang ◽  
Jie Chen
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
High Efficiency ◽  
Theory And Practice ◽  
Long Time ◽  
And Performance ◽  
Binary Tree Search ◽  
The One ◽  
Sparse Fast Fourier Transform ◽  
L1 And L2

Computing the sparse fast Fourier transform (sFFT) has emerged as a critical topic for a long time because of its high efficiency and wide practicability. More than twenty different sFFT algorithms compute discrete Fourier transform (DFT) by their unique methods so far. In order to use them properly, the urgent topic of great concern is how to analyze and evaluate the performance of these algorithms in theory and practice. This paper mainly discusses the technology and performance of sFFT algorithms using the aliasing filter. In the first part, the paper introduces the three frameworks: the one-shot framework based on the compressed sensing (CS) solver, the peeling framework based on the bipartite graph and the iterative framework based on the binary tree search. Then, we obtain the conclusion of the performance of six corresponding algorithms: the sFFT-DT1.0, sFFT-DT2.0, sFFT-DT3.0, FFAST, R-FFAST, and DSFFT algorithms in theory. In the second part, we make two categories of experiments for computing the signals of different SNRs, different lengths, and different sparsities by a standard testing platform and record the run time, the percentage of the signal sampled, and the L0, L1, and L2 errors both in the exactly sparse case and the general sparse case. The results of these performance analyses are our guide to optimize these algorithms and use them selectively.

Indentation of a Transversely Isotropic Thermoporoelastic Half-Space by a Rigid Circular Cylindrical Punch

2017 ◽  
Vol 84 (11) ◽  
Author(s):  
Yilan Huang ◽  
Guozhan Xia ◽  
Weiqiu Chen ◽  
Xiangyu Li
Keyword(s):  
Half Space ◽  
Original Problem ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
The Finite Element Method ◽  
Thermal Fields ◽  
Cylindrical Punch ◽  
The One ◽  
Physical Quantities ◽  
Potential Theory Method

Exact solutions to the three-dimensional (3D) contact problem of a rigid flat-ended circular cylindrical indenter punching onto a transversely isotropic thermoporoelastic half-space are presented. The couplings among the elastic, hydrostatic, and thermal fields are considered, and two different sets of boundary conditions are formulated for two different cases. We use a concise general solution to represent all the field variables in terms of potential functions and transform the original problem to the one that is mathematically expressed by integral (or integro-differential) equations. The potential theory method is extended and applied to exactly solve these integral equations. As a consequence, all the physical quantities of the coupling fields are derived analytically. To validate the analytical solutions, we also simulate the contact behavior by using the finite element method (FEM). An excellent agreement between the analytical predictions and the numerical simulations is obtained. Further attention is also paid to the discussion on the obtained results. The present solutions can be used as a theoretical reference when practically applying microscale image formation techniques such as thermal scanning probe microscopy (SPM) and electrochemical strain microscopy (ESM).

Note on the FFT Based Computational Code and Its Application

2008 ◽  
Author(s):  
Xiaoqing Jin ◽  
Leon M. Keer ◽  
Qian Wang
Keyword(s):  
Numerical Simulation ◽  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Toeplitz Matrix ◽  
Contact Problems ◽  
Point Of View ◽  
Fast Fourier Transform Algorithm ◽  
Discrete Convolution ◽  
Mathematical Point ◽  
Computational Code

The discrete convolution based Fast Fourier Transform algorithm (DC-FFT) has been successfully applied in numerical simulation of contact problems. The algorithm is revisited from a mathematical point of view, equivalent to a Toeplitz matrix multiplied by a vector. The nature of the convolution property permits one to implement the algorithm with fewer constraints in choosing the computational domains. This advantageous feature is explored in the present work, and is expected to be beneficial to many tribological studies.

An Elastohydrodynamic Lubrication Model for Coated Surfaces in Point Contacts

2007 ◽  
Vol 129 (3) ◽  
pp. 509-516 ◽  
Author(s):  
Yuchuan Liu ◽  
W. Wayne Chen ◽  
Dong Zhu ◽  
Shuangbiao Liu ◽  
Q. Jane Wang
Keyword(s):  
Fourier Transform ◽  
Film Thickness ◽  
Fast Fourier Transform ◽  
Elastic Deformation ◽  
Elastohydrodynamic Lubrication ◽  
Contact Radius ◽  
Point Contacts ◽  
Compliant Coating ◽  
Influence Coefficients ◽  
Coated Surfaces

An elastohydrodynamic lubrication (EHL) model for coated surfaces in point contacts has been developed by combining the elastic deformation formulation for the coated surfaces with an EHL model. Inverse fast Fourier transform (IFFT) is employed first to obtain the influence coefficients (ICs) from the frequency response function (FRF). The subsequent calculation of elastic deformation is performed using the efficient algorithm of discrete convolution and fast Fourier transform (DC-FFT). The coating EHL model is verified by the comparison to available numerical results. The effects of coating on lubrication under various loads, speeds, rheological models, and pressure-viscosity behaviors are numerically investigated. Similar to the observations from dry contact, stiffer coatings in EHL tend to reduce the nominal contact radius but increase the maximum contact pressure, and vice versa for more compliant coatings. However, as coating thickness increases, the influence of coatings on film thickness, including the central and the minimum film thicknesses, does not follow a monotonic variation, and therefore, cannot be predicted by any simple film thickness equation. The reason for that is the pressure viscosity effect which tends to counterbalance the effect of coating. The average friction coefficient in lubricant film increases in stiff coating cases but decreases for compliant coating cases. Furthermore, two possible approaches to improving the minimum film thickness thus reducing friction and wear in mixed lubrication are indicated: a thin stiff coating for conventional EHL and a thick compliant coating for soft EHL.

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