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2008a): A novel time integration method for solving a large system of nonlinear algebraic equations
 CMES: Computer Modeling in Engineering & Sciences
"... Abstract: Iterative algorithms for solving a nonlinear system of algebraic equations of the type: Fi(x j) = 0, i, j = 1,...,n date back to the seminal work of Issac Newton. Nowadays a Newtonlike algorithm is still the most popular one due to its easy numerical implementation. However, this type ..."
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Cited by 20 (15 self)
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Abstract: Iterative algorithms for solving a nonlinear system of algebraic equations of the type: Fi(x j) = 0, i, j = 1,...,n date back to the seminal work of Issac Newton. Nowadays a Newtonlike algorithm is still the most popular one due to its easy numerical implementation. However, this type of algorithm is sensitive to the initial guess of the solution and is expensive in the computations of the Jacobian matrix ∂Fi/∂x j and its inverse at each iterative step. In a timeintegration of a system of nonlinear Ordinary Differential Equations (ODEs) of the type Bi jx ̇ j + Fi = 0 where Bi j are nonlinear functions of x j, the methods which involve an inverse of the Jacobain matrix Bi j = ∂Fi/∂x j are called “Implicit”, while those that do not involve an inverse of ∂Fi/∂x j are called “Explicit”. In this paper a natural system of explicit ODEs is derived from the given system of nonlinear algebraic equations (NAEs), by introducing a fictitious time, such that it is a mathematically equivalent system in the n+1dimensional space as the original algebraic equations system is in the ndimensional space. The iterative equations are obtained by applying numerical integrations on the resultant ODEs, which do not need the information of ∂Fi/∂x j and its inverse. The computational cost is thus greatly reduced. Numerical examples given confirm that this fictitious time integration method (FTIM) is highly efficient to find the true solutions with residual errors being much smaller. Also, the FTIM is used to study the attracting sets of fixed points, when multiple roots exist.
A fictitious time integration method for the numerical solution of the Fredholm integral equation and for numerical differentiation of noisy data, and its relation to the filter theory. CMES: Computer Modeling in Engineering
 Sciences
, 2009
"... Abstract: The Fictitious Time Integration Method (FTIM) previously developed by Liu and Atluri (2008a) is employed here to solve a system of illposed linear algebraic equations, which may result from the discretization of a firstkind linear Fredholm integral equation. We rationalize the mathemati ..."
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Cited by 12 (12 self)
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Abstract: The Fictitious Time Integration Method (FTIM) previously developed by Liu and Atluri (2008a) is employed here to solve a system of illposed linear algebraic equations, which may result from the discretization of a firstkind linear Fredholm integral equation. We rationalize the mathematical foundation of the FTIM by relating it to the wellknown filter theory. For the linear ordinary differential equations which are obtained through the FTIM (and which are equivalently used in FTIM to solve the illposed linear algebraic equations), we find that the fictitous time plays the role of a regularization parameter, and its filtering effect is better than that of the Tikhonov and the exponential filters. Then, we apply this new method to solve the problem of numerical differentiation of noisy data [such as finding da/dN in fatigue, where a is the measured cracklength and N is the number of load cycles], and the inversion of the Abel integral equation under noise. It is established that the numerical method of FTIM is robust against the noise.
TTrefftz Voronoi cell finite elements with elastic/rigid inclusions or voids for micromechanical analysis of composite and porous materials
, 2012
"... In this paper, we develop TTrefftz Voronoi Cell Finite Elements (VCFEMTTs) for micromechanical modeling of composite and porous materials. In addition to a homogenous matrix in each polygonshaped element, three types of arbitrarilyshaped heterogeneities are considered in each element: an elast ..."
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Cited by 12 (11 self)
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In this paper, we develop TTrefftz Voronoi Cell Finite Elements (VCFEMTTs) for micromechanical modeling of composite and porous materials. In addition to a homogenous matrix in each polygonshaped element, three types of arbitrarilyshaped heterogeneities are considered in each element: an elastic inclusion, a rigid inclusion, or a void. In all of these three cases, an interelement compatible displacement field is assumed along the element outerboundary, and interior displacement fields in the matrix as well as in the inclusion are independently assumed as TTrefftz trial functions. Characteristic lengths are used for each element to scale the TTrefftz trial functions, in order to avoid solving systems of illconditioned equations. Two approaches for developing element stiffness matrices are used. The differences between these two approaches are that, the compatibility between the independently assumed fields at the outer as well as the innerboundary, are enforced alternatively, by Lagrange multipliers in multifield boundary variational principles, or by collocation at a finite number of preselected points. Following a previous paper of the authors, these elements are denoted as VCFEM
A simple multisourcepoint trefftz method for solving direct/inverse shm problems of plane elasticity in arbitrary multiplyconnected domains
 CMES: Computer Modeling in Engineering & Sciences
, 2012
"... Abstract: In this paper, a generalized Trefftz method in plane elasticity is developed, for solving problems in an arbitrary multiply connected domain. Firstly, the relations between Trefftz basis functions from different source points are discussed, by using the binomial theorem and the logarithmi ..."
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Cited by 10 (10 self)
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Abstract: In this paper, a generalized Trefftz method in plane elasticity is developed, for solving problems in an arbitrary multiply connected domain. Firstly, the relations between Trefftz basis functions from different source points are discussed, by using the binomial theorem and the logarithmic binomial theorem. Based on these theorems, we clearly explain the relation between the TTrefftz and the FTrefftz methods, and why the traditional TTrefftz method, which uses only one source point, cannot successfully solve problems in a multiply connected domain with genus larger than 1. Thereafter, a generalized Trefftz method is proposed, which uses logarithmic and negative power series from multiple source points, and positive power series from only one source point, as complex potentials. In addition, a characteristic length for each source point is used to scale the Trefftz basis functions, in order to resolve the illposedness of Trefftz methods. For direct problems, no further regularization techniques are used, because the coefficient matrix of the system of linear equations to be solved is already wellconditioned, by using characteristic lengths to scale the Trefftz basis functions. Inverse problems in plane
A scalar homotopy method for solving an over/under determined system of nonlinear algebraic equations
 CMES: Computer Modeling in Engineering and Sciences
, 2009
"... Abstract: Iterative algorithms for solving a system of nonlinear algebraic equations (NAEs): Fi(x j) = 0, i, j = 1,...,n date back to the seminal work of Issac Newton. Nowadays a Newtonlike algorithm is still the most popular one to solve the NAEs, due to the ease of its numerical implementation ..."
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Cited by 10 (7 self)
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Abstract: Iterative algorithms for solving a system of nonlinear algebraic equations (NAEs): Fi(x j) = 0, i, j = 1,...,n date back to the seminal work of Issac Newton. Nowadays a Newtonlike algorithm is still the most popular one to solve the NAEs, due to the ease of its numerical implementation. However, this type of algorithm is sensitive to the initial guess of solution, and is expensive in terms of the computations of the Jacobian matrix ∂Fi/∂x j and its inverse at each iterative step. In addition, the Newtonlike methods restrict one to construct an iteration procedure for nvariables by using nequations, which is not a necessary condition for the existence of a solution for underdetermined or overdetermined system of equations. In this paper, a natural system of firstorder nonlinear Ordinary Differential Equations (ODEs) is derived from the given system of Nonlinear Algebraic Equations (NAEs), by introducing a scalar homotopy function gauging the total residual error of the system of equations. The iterative equations are obtained by numerically integrating the resultant ODEs, which does not need the inverse of ∂Fi/∂x j. The new method keeps the merit of homotopy method, such as the global convergence, but it does not involve the complicated computation of the inverse of the Jacobian matrix. Numerical examples given confirm that this Scalar Homotopy Method (SHM) is highly efficient to find the true solutions with residual errors being much smaller.
Development of 3D Trefftz Voronoi Cells with/without Spherical Voids &/or Elastic/Rigid Inclusions for Micromechanical Modeling of Heterogeneous Materials
 CMC: COMPUTERS, MATERIALS & CONTINUA
, 2012
"... In this paper, as an extension to the authors’s work in [Dong and Atluri ..."
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Cited by 9 (8 self)
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In this paper, as an extension to the authors’s work in [Dong and Atluri
Development of 3D TTrefftz Voronoi Cell Finite Elements with/without Spherical Voids & /or Elastic/Rigid Inclusions for Micromechanical Modeling of Heterogeneous Materials
 CMC: COMPUTERS, MATERIALS & CONTINUA
, 2012
"... In this paper, threedimensional TTrefftz Voronoi Cell Finite Elements (VCFEMTTs) are developed for micromechanical modeling of heterogeneous materials. Several types of VCFEMs are developed, depending on the types of heterogeneity in each element. Each VCFEM can include alternatively a spherical ..."
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Cited by 8 (7 self)
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In this paper, threedimensional TTrefftz Voronoi Cell Finite Elements (VCFEMTTs) are developed for micromechanical modeling of heterogeneous materials. Several types of VCFEMs are developed, depending on the types of heterogeneity in each element. Each VCFEM can include alternatively a spherical void, a spherical elastic inclusion, a spherical rigid inclusion, or no voids/inclusions at all.In all of these cases, an interelement compatible displacement field is assumed at each surface of the polyhedral element, with Barycentric coordinates as nodal shape functions.The TTrefftz trial displacement fields in each element are expressed in terms of the PapkovichNeuber solution. Spherical harmonics are used as the PapkovichNeuber potentials to derive the TTrefftz trial displacement fields. Characteristic lengths are used for each element to scale the TTrefftz trial functions, in order to avoid solving systems of illconditioned equations. Two approaches for developing element stiffness matrices are used. The differences between these two approachesare that, the compatibility between the independently assumed fields in the interior of the element with those at the outer as well as the innerboundary, are enforced alternatively, by Lagrange multipliers in multifield boundary varia
Large Deformation Analyses of SpaceFrame Structures, Using Explicit Tangent Stiffness Matrices, Based on the Reissner variational principle and a von Karman Type Nonlinear Theory in Rotated Reference Frames
"... Abstract: This paper presents a simple finite element method, based on assumed moments and rotations, for geometrically nonlinear large rotation analyses of space frames consisting of members of arbitrary crosssection. A von Karman type nonlinear theory of deformation is employed in the updated La ..."
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Cited by 7 (7 self)
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Abstract: This paper presents a simple finite element method, based on assumed moments and rotations, for geometrically nonlinear large rotation analyses of space frames consisting of members of arbitrary crosssection. A von Karman type nonlinear theory of deformation is employed in the updated Lagrangian corotational reference frame of each beam element, to account for bending, stretching, and torsion of each element. The Reissner variational principle is used in the updated Lagrangian corotational reference frame, to derive an explicit expression for the (12x12) symmetric tangent stiffness matrix of the beam element in the corotational reference frame. The explicit expression for the finite rotation of the axes of the corotational reference frame, from the global Cartesian reference frame is derived from the finite displacement vectors of the 2 nodes of each finite element. Thus, the explicit expressions for the tangent stiffness matrix of each finite element of the beam, in the global Cartesian frame, can be seen to be derived as textbook examples of nonlinear analyses. When compared to the primal (displacement) approach wherein C1 continuous trial functions (for transverse displacements) over each el
A highly accurate technique for interpolations using very highorder polynomials, and its applications to some illposed linear problems
 CMES: Computer Modeling in Engineering & Sciences
, 2009
"... Abstract: Since the works of Newton and Lagrange, interpolation had been a mature technique in the numerical mathematics. Among the many interpolation methods, global or piecewise, the polynomial interpolation p(x) = a0+a1x+...+ anx n expanded by the monomials is the simplest one, which is easy to ..."
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Cited by 6 (5 self)
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Abstract: Since the works of Newton and Lagrange, interpolation had been a mature technique in the numerical mathematics. Among the many interpolation methods, global or piecewise, the polynomial interpolation p(x) = a0+a1x+...+ anx n expanded by the monomials is the simplest one, which is easy to handle mathematically. For higher accuracy, one always attempts to use a higherorder polynomial as an interpolant. But, Runge gave a counterexample, demonstrating that the polynomial interpolation problem may be illposed. Very highorder polynomial interpolation is very hard to realize by numerical computations. In this paper we propose a new polynomial interpolation by p(x) = ā0 + ā1x/R0 +...+ ānxn/Rn0, where R0 is a characteristic length used as a parameter, and chosen by the user. The resulting linear equations system to solve the coefficients āα is wellconditioned, if a suitable R0 is chosen. We define a nondimensional parameter, R∗0 = R0/(b−a) [where a and b are the endpoints of the interval for x]. The range of values for R∗0 for numerical stability is identified, and one can overcome the difficulty due to
Development of TTrefftz fournode quadrilateral and Voronoi Cell Finite Elements for macro & micromechanical modeling of solids
 CMES: COMPUTER MODELING IN ENGINEERING & SCIENCES
, 2011
"... In this paper, we explore three different ways of developing TTrefftz finite elements of quadrilateral as well as polygonal shapes. In all of these three approaches, in addition to assuming an interelement compatible displacement field along the element boundary, an interior displacement field fo ..."
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Cited by 5 (5 self)
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In this paper, we explore three different ways of developing TTrefftz finite elements of quadrilateral as well as polygonal shapes. In all of these three approaches, in addition to assuming an interelement compatible displacement field along the element boundary, an interior displacement field for each element is independently assumed as a linear combination of TTrefftz trial functions. In addition, a characteristic length is defined for each element to scale the TTrefftz modes, in order to avoid solving systems of illconditioned equations. The differences between these three approaches are that, the compatibility between the independently assumed fields at the boundary and in the interior, are enforced alternatively, using a twofield boundary variational principle, collocation, and the least squares method. The corresponding fournode quadrilateral elements with/without drilling degrees of freedom are developed, for modeling macrostructures of solids. These three approaches are also used to derive TTrefftz Voronoi Cell Finite Elements (VCFEM), for micromechanical analysis of heterogeneous materials. Several two dimensional macro & micromechanical problems are solved using these elements. Computational results demonstrate that the elements derived using the collocation method are very simple, accurate and computationally efficient. Because the elements derived by this approach are also not plagued by LBB conditions, which are almost impossible to be satisfied a priori, we consider this class of elements to be useful for engineering applications in micromechanical modeling of heterogeneous materials.