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A globally optimal iterative algorithm using the best descent vector ẋ = λ[αcF+ BT F], with the critical value αc, for solving a system of nonlinear algebraic equations F(x)= 0”, CMES
"... Abstract: An iterative algorithm based on the concept of best descent vector u in x ̇ = λu is proposed to solve a system of nonlinear algebraic equations (NAEs): F(x) = 0. In terms of the residual vector F and a monotonically increasing positive function Q(t) of a timelike variable t, we define ..."
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Cited by 5 (3 self)
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Abstract: An iterative algorithm based on the concept of best descent vector u in x ̇ = λu is proposed to solve a system of nonlinear algebraic equations (NAEs): F(x) = 0. In terms of the residual vector F and a monotonically increasing positive function Q(t) of a timelike variable t, we define a future cone in the Minkowski space, wherein the discrete dynamics of the proposed algorithm evolves. A new method to approximate the best descent vector is developed, and we find a critical value of the weighting parameter αc in the best descent vector u = αcF+BTF, where B = ∂F/∂x is the Jacobian matrix. We can prove that such an algorithm leads to the largest convergence rate with the descent vector given by u = αcF+BTF; hence we label the present algorithm as a globally optimal iterative algorithm (GOIA). Some numerical examples are used to validate the performance of the GOIA; a very fast convergence rate in finding the solution is observed.
SOLUTIONS OF THE VON KÁRMÁN PLATE EQUATIONS BY A GALERKIN METHOD, WITHOUT INVERTING THE TANGENT STIFFNESS MATRIX
, 2014
"... Large deflections of a simply supported von Kármán plate with imperfect initial deflections, under a combination of inplane loads and lateral pressure, are analyzed by a semianalytical global Galerkin method. While many may argue that the dominance of the finite element method in the marketplace ma ..."
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Cited by 3 (2 self)
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Large deflections of a simply supported von Kármán plate with imperfect initial deflections, under a combination of inplane loads and lateral pressure, are analyzed by a semianalytical global Galerkin method. While many may argue that the dominance of the finite element method in the marketplace may make any other attempts to solve nonlinear plate problems to be redundant and obsolete, semi and precise analytical methods, when possible, simply serve as benchmark solutions if nothing else. Also, since parametric variations are simpler to access through such analytical methods, they are more useful in studying the physics of the phenomena. In the present method, the Galerkin scheme is first applied to transform the governing nonlinear partial differential equations of the von Kármán plate into a system of general nonlinear algebraic equations (NAEs) in an explicit form. The Jacobian matrix, the tangent stiffness matrix of the system of NAEs, is explicitly derived, which speeds up the Newton–Raphson iterative method if it is used. The present global Galerkin method is compared with the incremental Galerkin method, the perturbation method, the finite element method and the finite difference method in solving the von Kármán plate equations to compare their relative accuracies and efficiencies. Buckling behavior and jump phenomenon of the plate are detected and analyzed. Besides the classical Newton– Raphson method, an entirely novel series of scalar homotopy methods, which do not need to invert the
NOVEL COMPUTATIONAL AND ANALYTIC TECHNIQUES FOR NONLINEAR SYSTEMS APPLIED TO STRUCTURAL AND CELESTIAL MECHANICS
, 2015
"... In this Dissertation, computational and analytic methods are presented to address nonlinear systems with applications in structural and celestial mechanics. Scalar Homotopy Methods (SHM) are first introduced for the solution of general systems of nonlinear algebraic equations. The methods are appli ..."
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In this Dissertation, computational and analytic methods are presented to address nonlinear systems with applications in structural and celestial mechanics. Scalar Homotopy Methods (SHM) are first introduced for the solution of general systems of nonlinear algebraic equations. The methods are applied to the solution of postbuckling and limit load problems of solids and structures as exemplified by simple plane elastic frames, considering only geometrical nonlinearities. In many problems, instead of simply adopting a root solving method, it is useful to study the particular problem in more detail in order to establish an especially efficient and robust method. Such a problem arises in satellite geodesy coordinate transformation where a new highly efficient solution, providing global accuracy with a noniterative sequence of calculations, is developed. Simulation results are presented to compare the solution accuracy and algorithm performance for applications spanning the LEOtoGEO range of missions. Analytic methods are introduced to address problems in structural mechanics and astrodynamics. Analytic transfer functions are developed
A Further Study on Using ẋ = λ [αR+βP] (P = F−R(F ·R)/‖R‖2) and ẋ = λ [αF+βP∗] (P ∗ = R−F(F ·R)/‖F‖2) in Iteratively Solving the Nonlinear System of Algebraic Equations F(x) = 0
"... Abstract: In this continuation of a series of our earlier papers, we define a hypersurface h(x, t) = 0 in terms of the unknown vector x, and a monotonically increasing function Q(t) of a timelike variable t, to solve a system of nonlinear algebraic equations F(x) = 0. If R is a vector related t ..."
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Abstract: In this continuation of a series of our earlier papers, we define a hypersurface h(x, t) = 0 in terms of the unknown vector x, and a monotonically increasing function Q(t) of a timelike variable t, to solve a system of nonlinear algebraic equations F(x) = 0. If R is a vector related to ∂h/∂x, we consider the evolution equation ẋ = λ [αR+βP], where P = F−R(F ·R)/‖R‖2 such that P ·R = 0; or x ̇ = λ [αF+ βP∗], where P ∗ = R−F(F ·R)/‖F‖2 such that P ∗ ·F = 0. From these evolution equations, we derive Optimal Iterative Algorithms (OIAs) with Optimal Descent Vectors (ODVs), abbreviated as ODV(R) and ODV(F), by deriving optimal values of α and β for fastest convergence. Several numerical examples illustrate that the present algorithms converge very fast. We also provide a solution of the nonlinear Duffing oscillator, by using a harmonic balance method and a postconditioner, when very highorder harmonics are considered.