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Fatigue Crack Growth Reliability Analysis by Stochastic Boundary Element Method
Get PDFIn this paper, a stochastic dual boundary element formulation is presented for probabilistic analysis of fatigue crack growth. The method involves a direct differentiation approach for calculating boundary and fracture response derivatives with respect to random parameters. Total derivatives method is used to obtain the derivatives of fatigue parameters with respect to random parameters. First- Order Reliability Method (FORM) is applied to evaluate the most probable point (MPP). Opening mode fatigue crack growth problems are used as benchmarks to demonstrate the performance of the proposed method
Analysis of 3D Anisotropic Solids Using Fundamental Solutions Based on Fourier Series and the Adaptive Cross Approximation Method
Get PDFThe efficient evaluation of the fundamental solution for 3D general anisotropic elasticity is a subject of great interest in the BEM community due to its mathematical complexity. Recently, Tan, Shiah, andWang (2013) have represented the algebraically explicit form of it developed by Ting and Lee (Ting and Lee, 1997; Lee, 2003) by a computational efficient double Fourier series. The Fourier coefficients are numerically evaluated only once for a specific material and are independent of the number of field points in the BEM analysis. This work deals with the application of hierarchical matrices and low rank approximations, applying the Adaptive Cross Approximation (ACA) to treat 3D general anisotropic solids in BEM using this Green’s function based on Fourier series. The use of hierarchical format is aimed at reducing the storage requirements of the system matrices and the computational effort in the BEM analysis of large systems. Numerical examples are presented to show the successful implementation of using ACA and the formulation based on Fourier series for BEM analysis of 3D anisotropic solids
Solution of Two-dimensional Linear and Nonlinear Unsteady Schrödinger Equation using “Quantum Hydrodynamics” Formulation with a MLPG Collocation Method
Get PDFA numerical solution of the linear and nonlinear time-dependent Schrödinger equation is obtained, using the strong form MLPG Collocation method. Schrödinger equation is replaced by a system of coupled partial differential equations in terms of particle density and velocity potential, by separating the real and imaginary parts of a general solution, called a quantum hydrodynamic (QHD) equation, which is formally analogous to the equations of irrotational motion in a classical fluid. The approximation of the field variables is obtained with the Moving Least Squares (MLS) approximation and the implicit Crank-Nicolson scheme is used for time discretization. For the two-dimensional nonlinear Schrödinger equation, the lagging of coefficients method has been utilized to eliminate the nonlinearity of the corresponding examined problem. A Type-I nodal distribution is used in order to provide convergence for the discrete Laplacian operator used at the governing equation. Numerical results are validated, comparing them with analytical and numerical solutions
Image Segmentation Method for Complex Vehicle Lights Based on Adaptive Significance Level Set
Get PDFThe existing study on the image segmentation methods based on the image of vehicle lights is insufficient both at home and abroad, and its segmentation efficiency and accuracy is low as well. On the basis of the analysis of the regional characteristics of vehicle lights and a level set model, an image segmentation method for complex vehicle lights based on adaptive significance level set contour model is proposed in this paper. Adaptive positioning algorithm of the significant initial contour curve based on two-dimensional convex hull is designed to obtain the initial position of evolution curve, thus the adaptive ability of the model is improved. Meanwhile, in order to solve the problem which the image edges are blurred when Gaussian filter is used to remove image noise in Li model, the regularized P-M equation is adopted to achieve effective maintenance of image edge information while the noise is effectively removed. Experimental results show that the image segmentation accuracy for different lights is up to around 95%, the proposed method can significantly reduce the number of iterations and improve segmentation efficiency, which have the advantages of higher accuracy and fast speed, and it can provide a strong support for the accurate vehicle recognition
Eshelby Stress Tensor T: a Variety of Conservation Laws for T in Finite Deformation Anisotropic Hyperelastic Solid & Defect Mechanics, and the MLPG-Eshelby Method in Computational Finite Deformation Solid Mechanics-Part I
Get PDFThe concept of a stress tensor [for instance, the Cauchy stress σ, Cauchy (1789-1857); the first Piola-Kirchhoff stress P, Piola (1794-1850), and Kirchhoff (1824-1889); and the second Piola-Kirchhoff stress, S] plays a central role in Newtonian continuum mechanics, through a physical approach based on the conservation laws for linear and angular momenta. The pioneering work of Noether (1882-1935), and the extraordinarily seminal work of Eshelby (1916- 1981), lead to the concept of an “energy-momentum tensor” [Eshelby (1951)]. An alternate form of the “energy-momentum tensor” was also given by Eshelby (1975) by taking the two-point deformation gradient tensor as an independent field variable; and this leads to a stress measure T (which may be named as the Eshelby Stress Tensor). The corresponding conservation laws for T in terms of the pathindependent integrals, given by Eshelby (1975), were obtained through a sequence of imagined operations to “cut the stress states” in the current configuration. These imagined operations can not conceptually be extended to nonlinear steady state or transient dynamic problems [Eshelby (1975)]. To the authors’ knowledge, these path-independent integrals for dynamic finite-deformations of inhomogeneous materials were first derived by Atluri (1982) by examining the various internal and external work quantities during finite elasto-visco-plastic dynamic deformations, to derive the energy conservation laws, in the undeformed configuration [ref. to Eq. (18) in Atluri (1982)]. The stress tensor T was derived, independently, in its path-independent integral form for computational purposes [ref. to Eq. (30) in Atluri (1982)]. The corresponding integrals were successfully applied to nonlinear dynamic fracture analysis to determine “the energy change rate”, denoted as T*. A similar analytical work for elasto-statics was reported by Hill (1986). With the use of the stress measure T for finite-deformation solid and defect mechanics, the concept of “the strength of the singularities”, labeled in this paper as the vector T*, is formulated for a defective hyperelastic anisotropic solid undergoing finite deformations, in its various path-independent integral forms.
We first derive a vector balance law for the Eshelby stress tensor T, and show that it involves a mathematically “weak-form” of a vector momentum balance law for P. In small deformation linear elasticity (where P, S and σ are all equivalent), the stress tensor σ is linear in the deformation gradient F. Even in small deformation linear elasticity, the Eshelby Stress Tensor T is quadratic in F. By considering the various weak-forms of the balance law for T itself, we derive a variety of “conservation laws” for T in Section 2. We derive four important “path-independent” integrals, TK∗, TL∗(L) , T∗(M), TIJ∗(G) , in addition to many others. We show the relation of TK∗, TL∗(L) , T∗(M) integrals to the J-, L- and M- integrals given in Knowles and Sternberg (1972). The four laws derived in this paper are, however, valid for finite-deformation anisotropic hyperelastic solid- and defect-mechanics. Some discussions related to the use of T in general computational solid mechanics of finitely deformed solids are given in Section 3. The application of the Eshelby stress tensor in computing the deformation of a one-dimensional bar is formulated in Section 4 for illustration purposes. We present two computational approaches: the Primal Meshless Local Petrov Galerkin (MLPG)-Eshelby Method, and the Mixed MLPG-Eshelby Method, as applications of the original MLPG method proposed by Atluri (1998,2004). More general applications of T directly, in computational solid mechanics of finitely deformed solids, will be reported in our forthcoming papers, for mechanical problems, in their explicitly-linearized forms, through the Primal MLPG-Eshelby and the Mixed MLPG-Eshelby Methods
Enrichment Procedures for Soft Clusters: A Statistical Test and its Applications
Get PDFClusters, typically mined by modeling locality of attribute spaces, are often evaluated for their ability to demonstrate ‘enrichment’ of categorical features. A cluster enrichment procedure evaluates the membership of a cluster for significant representation in predefined categories of interest. While classical enrichment procedures assume a hard clustering definition, this paper introduces a new statistical test that computes enrichments for soft clusters. Application of the new test to several scientific datasets is given
Disclosing the Complexity of Nonlinear Ship Rolling and Duffing Oscillators by a Signum Function
Get PDFIn this paper we study the nonlinear dynamical system x·=f(x,t) from a newly developed theory, viewing the time-varying function of sign(||f||2||x||2− 2(f·x)2) = −sign(cos 2θ) as a key factor, where θ is the intersection angle between x and f. It together with sign(cos θ) can reveal the complexity of nonlinear Duffing oscillator and a quadratic ship rolling oscillator. The barcode is formed by plotting sign(||f||2||x||2− 2(f·x)2) with respect to time. We analyze the barcode to point out the bifurcation of subharmonic motions and the range of chaos in the parameter space. The bifurcation diagram obtained by plotting the percentage of the first set of dis-connectivity A1− : = {sign(cos θ) = + 1 and sign(cos 2θ) = + 1} with respect to the amplitude of harmonic loading leads to a finer structure of a devil staircase for the ship rolling oscillator, as well as a cascade of subharmonic motions to chaos for the Duffing oscillator
Interval Uncertain Optimization of Vehicle Suspension for Ride Comfort
Get PDFBased on the interval analysis method, this paper proposes an uncertain optimization model for the ride comfort in vehicles and achieves the optimal design of vehicle ride comfort under the condition of complicated uncertainty. The spring stiffness and shock absorber damping of suspension is regarded as the design parameters, while the root mean square (RMS) of the vehicle body acceleration is treated as the design objective and the corresponding constraints are composed of suspension stiffness, natural frequency and RMS of suspension dynamic deflection. Moreover, the uncertainties of key parameters, such as sprung mass, tire stiffness, vehicle speed and road roughness, are also considered and quantified by interval analysis method. After that, an interval uncertain optimization model of vehicle suspension is established for ride comfort, which is subsequently converted to an ordinary deterministic optimization problem through a transformation model. Finally, the proposed method is applied to three typical vehicle suspension dynamic systems with two degrees of freedom, four degrees of freedom and seven degrees of freedom
Hydro-thermo-viscoelastic Based Finite Element Modeling of Apple Convective Drying Process
Get PDFIn the present work we aim to simulate unsteady two-dimensional evolution of the moisture content, temperature and mechanical stress in a parallelepiped apple sample during convective drying. The model is based on the heat and mass transfer equations and the mechanical equilibrium equation under the assumptions of plane deformation, viscoelasticity and isotropic hydric shrinkage. The Finite Elements COMSOL Multiphysics solver is used to solve the developed model. The hydro-thermal model was validated on experimental data drawn in our laboratory for moisture and temperature internal profiles of the product. Excellent agreement has been obtained between numerical and measured data for different drying temperatures. The hydro-thermal model was, then, implemented in a Finite Element hydro-thermo-viscoelastic developed model. Results for mechanical stresses are found to be in good agreement with former observations. It was found, also, that the maximum stress arises at the beginning of drying and holds at the top external surface of the sample which is the surface facing the airflow. The phenomenon is augmented by temperature
Computational Methods in Engineering: A Variety of Primal & Mixed Methods, with Global & Local Interpolations, for Well-Posed or Ill-Posed BCs
Get PDFIn this expository article, a variety of computational methods, such as Collocation, Finite Volume, Finite Element, Boundary Element, MLPG (Meshless Local Petrov Galerkin), Trefftz methods, and Method of Fundamental Solutions, etc., which are often used in isolated ways in contemporary literature are presented in a unified way, and are illustrated to solve a 4th order ordinary differential equation (beam on an elastic foundation). Both the primal formulation, which considers the 4th order ODE with displacement as the primitive variable, as well as two types of mixed formulations (one resulting in a set of 2 second-order ODEs, and the other resulting in a set of 4 first-order ODEs), which consider both displacement and its derivatives as mixed variables, are used as strong forms of the problem. Through integration by parts of the weighted residuals, different global and local, unsymmetric and symmetric weak-forms are derived. Both global (harmonics, polynomials, Radial Basis Functions, Trefftz and Fundamental solutions), and local interpolations (element-based interpolations, meshless Moving Least Squares) are used as trial functions of primal and mixed variables. By using Dirac Delta function, Heaviside function, Galerkin and Petrov Galerkin type of function, as well as fundamental solutions as test functions of various weak-forms, primal and mixed implementations of Collocation, Finite Volume, Finite Element, Boundary Element, Meshless Local Petrov Galerkin (MLPG), Trefftz and Method of Fundamental Solutions are developed. Applications of these methods are illustrated for solving problems with well-posed boundary conditions (BCs), which are the physically-consistent BCs of a solid-body (beam on elastic foundation), as well as for ill-posed boundary conditions, where the Cauchy type of B.C. are over-prescribed on a part of the boundary. The advantages & disadvantages of various primal & mixed, symmetric & unsymmetric weak forms are discussed, on the admissible order of continuity for trial & test functions, the requirement of evaluating higher-order differentials, as well as the enforcement of well-posed & ill-posed BCs. The relationship between various trial & test functions and the resulting sparse or dense, symmetric or non-symmetric, well-conditioned or ill-conditioned coefficient matrices are also demonstrated and discussed. This paper thus presents a unification of a variety of concepts in developing numerical methods for problems of multidisciplinary engineering and sciences, which are often presented in an ad hoc manner, in contemporary literature. The MATLAB codes pertaining to all the methods presented here are presented for free download at the website: www.care.eng.uci.edu/pubs.htm. This expository article will be a part of soon to be published introductory textbook by Atluri and Dong (2015)