Computer Methods in Applied Mechanics and Engineering
01 Jan 2010-
TL;DR: In this article, a cubic Hermite collocation scheme for the solution of the coupled integro-partial differential equations governing the propagation of a hydraulic fracture in a state of plane strain is described.
Abstract: article i nfo We describe a novel cubic Hermite collocation scheme for the solution of the coupled integro-partial differential equations governing the propagation of a hydraulic fracture in a state of plane strain. Special blended cubic Hermite-power-law basis functions, with arbitrary index 0b αb1, are developed to treat the singular behavior of the solution that typically occurs at the tips of a hydraulic fracture. The implementation of blended infinite elements to model semi-infinite crack problems is also described. Explicit formulae for the integrated kernels associated with the cubic Hermite and blended basis functions are provided. The cubic Hermite collocation algorithm is used to solve a number of different test problems with two distinct propagation regimes and the results are shown to converge to published similarity and asymptotic solutions. The convergence rate of the cubic Hermite scheme is determined by the order of accuracy of the tip asymptotic expansion as well as the O(h 4 ) error due to the Hermite cubic interpolation. The errors due to these two approximations need to be matched in order to achieve optimal convergence. Backward Euler time-stepping yields a robust algorithm that, along with geometric increments in the time-step, can be used to explore the transition between propagation regimes over many orders of magnitude in time.
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TL;DR: The review indicates that future researches should be oriented towards improving the efficiency of search techniques and approximation methods for large-scale building optimization problems; and reducing time and effort for such activities.
1,009 citations
Cites background from "Computer Methods in Applied Mechani..."
...Since then this research field has become a “fertile ground” for researchers as reviewed in references [103; 99]....
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TL;DR: T-splines, a generalization of NURBS enabling local refinement, have been explored as a basis for isogeometric analysis in this paper, and they have shown good results on some elementary two-dimensional and three-dimensional fluid and structural analysis problems and attain good results in all cases.
975 citations
Cites methods from "Computer Methods in Applied Mechani..."
...This concept is referred to as Isogeometric Analysis, and was first propos ed in Hughes, Cottrell, and Bazilevs [26] and further developed in [3, 4, 6, 7, 16, 17, 19, 20, 24, 42, 45]....
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TL;DR: A fully-coupled monolithic formulation of the fluid-structure interaction of an incompressible fluid on a moving domain with a nonlinear hyperelastic solid is presented.
Abstract: We present a fully-coupled monolithic formulation of the fluid-structure interaction of an incompressible fluid on a moving domain with a nonlinear hyperelastic solid. The arbitrary Lagrangian–Eulerian description is utilized for the fluid subdomain and the Lagrangian description is utilized for the solid subdomain. Particular attention is paid to the derivation of various forms of the conservation equations; the conservation properties of the semi-discrete and fully discretized systems; a unified presentation of the generalized-α time integration method for fluid-structure interaction; and the derivation of the tangent matrix, including the calculation of shape derivatives. A NURBS-based isogeometric analysis methodology is used for the spatial discretization and three numerical examples are presented which demonstrate the good behavior of the methodology.
866 citations
Cites background from "Computer Methods in Applied Mechani..."
..., [44, 45] for background) emerges as a key concept....
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...For a discussion of the import ance of conservation and satisfaction of the DGCL for moving domain problems see [17, 22, 44]....
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TL;DR: In this article, the half-point rule is introduced for NURBS-based isogeometric analysis, indicating that optimal rules involve a number of points roughly equal to half the number of degrees of freedom, or equivalently half the basis functions of the space under consideration.
486 citations
Cites background from "Computer Methods in Applied Mechani..."
...FEM discretization on a unifor m mesh and NURBS discretization on a uniform grid of control points are consi dered (see [13] for more details)....
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...I n particular, the upper part of the discrete spectrum is much better behaved [13], re sulting in better conditioned discrete systems....
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...In addi tion, on a per degree-offreedom basis, isogeometric analysis has exhibited superi or accuracy and robustness compared with finite element analysis (see [1, 5, 13])....
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...Com parison ofC andC discretizations has been thoroughly studied in [13], which we refer for an analysis of the results obtained by exact integration....
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TL;DR: This contribution provides a survey on approaches for performing Reliability-based Optimization, with emphasis on the theoretical foundations and the main assumptions involved.
Abstract: Reliability-based Optimization is a most appropriate and advantageous methodology for structural design. Its main feature is that it allows determining the best design solution (with respect to prescribed criteria) while explicitly considering the unavoidable effects of uncertainty. In general, the application of this methodology is numerically involved, as it implies the simultaneous solution of an optimization problem and also the use of specialized algorithms for quantifying the effects of uncertainties. In view of this fact, several approaches have been developed in the literature for applying this methodology in problems of practical interest. This contribution provides a survey on approaches for performing Reliability-based Optimization, with emphasis on the theoretical foundations and the main assumptions involved. Early approaches as well as the most recently developed methods are covered. In addition, a qualitative comparison is performed in order to provide some general guidelines on the range of applicability on the different approaches discussed in this contribution.
329 citations
Cites background or result from "Computer Methods in Applied Mechani..."
...According to results in the literature (see, e.g. Ching and Hsieh (2007a); Taflanidis and Beck (2008b); Jensen et al (2009)), the number of function evaluations required for solving a RBO problem may vary between three and five orders of magnitude....
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...In Jensen et al (2009), another approach for solving RBO problems based on sensitivity information is presented....
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References
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Book•
01 Jan 1970
17,608 citations
Book•
01 Jan 1967TL;DR: The dynamique des : fluides Reference Record created on 2005-11-18 is updated on 2016-08-08 and shows improvements in the quality of the data over the past decade.
Abstract: Preface Conventions and notation 1. The physical properties of fluids 2. Kinematics of the flow field 3. Equations governing the motion of a fluid 4. Flow of a uniform incompressible viscous fluid 5. Flow at large Reynolds number: effects of viscosity 6. Irrotational flow theory and its applications 7. Flow of effectively inviscid liquid with vorticity Appendices.
11,187 citations
Book•
01 Jan 1978
TL;DR: This book presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines as well as specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting.
Abstract: This book is based on the author's experience with calculations involving polynomial splines. It presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines. After two chapters summarizing polynomial approximation, a rigorous discussion of elementary spline theory is given involving linear, cubic and parabolic splines. The computational handling of piecewise polynomial functions (of one variable) of arbitrary order is the subject of chapters VII and VIII, while chapters IX, X, and XI are devoted to B-splines. The distances from splines with fixed and with variable knots is discussed in chapter XII. The remaining five chapters concern specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting. The present text version differs from the original in several respects. The book is now typeset (in plain TeX), the Fortran programs now make use of Fortran 77 features. The figures have been redrawn with the aid of Matlab, various errors have been corrected, and many more formal statements have been provided with proofs. Further, all formal statements and equations have been numbered by the same numbering system, to make it easier to find any particular item. A major change has occured in Chapters IX-XI where the B-spline theory is now developed directly from the recurrence relations without recourse to divided differences. This has brought in knot insertion as a powerful tool for providing simple proofs concerning the shape-preserving properties of the B-spline series.
10,258 citations
01 Jan 1988
TL;DR: In this article, the Riccati method is used to solve boundary value problems for Ordinary Differential Equations and to solve nonlinear problems for BVPSs in the standard form.
Abstract: List of Examples Preface 1. Introduction. Boundary Value Problems for Ordinary Differential Equations Boundary Value Problems in Applications 2. Review of Numerical Analysis and Mathematical Background. Errors in Computation Numerical Linear Algebra Nonlinear Equations Polynomial Interpolation Piecewise Polynomials, or Splines Numerical Quadrature Initial Value Ordinary Differential Equations Differential Operators and Their Discretizations 3. Theory of Ordinary Differential Equations. Existence and Uniqueness Results Green's Functions Stability of Initial Value Problems Conditioning of Boundary Value Problems 4. Initial Value Methods. Introduction. Shooting Superposition and Reduced Superposition Multiple Shooting for Linear Problems Marching Techniques for Multiple Shooting The Riccati Method Nonlinear Problems 5. Finite Difference Methods. Introduction Consistency, Stability, and Convergence Higher-Order One-Step Schemes Collocation Theory Acceleration Techniques Higher-Order ODEs Finite Element Methods 6. Decoupling. Decomposition of Vectors Decoupling of the ODE Decoupling of One-Step Recursions Practical Aspects of Consistency Closure and Its Implications 7. Solving Linear Equations. General Staircase Matrices and Condensation Algorithms for the Separated BC Case Stability for Block Methods Decomposition in the Nonseparated BC Case Solution in More General Cases 8. Solving Nonlinear Equations. Improving the Local Convergence of Newton's Method Reducing the Cost of the Newton Iteration Finding a Good Initial Guess Further Remarks on Discrete Nonlinear BVPS 9. Mesh Selection. Introduction Direct Methods A Mesh Strategy for Collocation Transformation Methods General Considerations 10. Singular Perturbations. Analytical Approaches Numerical Approaches Difference Methods Initial Value Methods 11. Special Topics. Reformulation of Problems in 'Standard' Form Generalized ODEs and Differential Algebraic Equations Eigenvalue Problems BVPs with Singularities Infinite Intervals Path Following, Singular Points and Bifurcation Highly Oscillatory Solutions Functional Differential Equations Method of Lines for PDEs Multipoint Problems On Code Design and Comparison Appendix A. A Multiple Shooting Code Appendix B. A Collocation Code References Bibliography Index.
1,210 citations
Book•
01 Jan 1985
TL;DR: This book discusses Consistency, Stability, and Convergence higher-Order One-Step Schemes Collocation Theory Acceleration Techniques Higher-Order ODEs Finite Element Methods and Initial Value Methods.
Abstract: List of Examples Preface 1. Introduction. Boundary Value Problems for Ordinary Differential Equations Boundary Value Problems in Applications 2. Review of Numerical Analysis and Mathematical Background. Errors in Computation Numerical Linear Algebra Nonlinear Equations Polynomial Interpolation Piecewise Polynomials, or Splines Numerical Quadrature Initial Value Ordinary Differential Equations Differential Operators and Their Discretizations 3. Theory of Ordinary Differential Equations. Existence and Uniqueness Results Green's Functions Stability of Initial Value Problems Conditioning of Boundary Value Problems 4. Initial Value Methods. Introduction. Shooting Superposition and Reduced Superposition Multiple Shooting for Linear Problems Marching Techniques for Multiple Shooting The Riccati Method Nonlinear Problems 5. Finite Difference Methods. Introduction Consistency, Stability, and Convergence Higher-Order One-Step Schemes Collocation Theory Acceleration Techniques Higher-Order ODEs Finite Element Methods 6. Decoupling. Decomposition of Vectors Decoupling of the ODE Decoupling of One-Step Recursions Practical Aspects of Consistency Closure and Its Implications 7. Solving Linear Equations. General Staircase Matrices and Condensation Algorithms for the Separated BC Case Stability for Block Methods Decomposition in the Nonseparated BC Case Solution in More General Cases 8. Solving Nonlinear Equations. Improving the Local Convergence of Newton's Method Reducing the Cost of the Newton Iteration Finding a Good Initial Guess Further Remarks on Discrete Nonlinear BVPS 9. Mesh Selection. Introduction Direct Methods A Mesh Strategy for Collocation Transformation Methods General Considerations 10. Singular Perturbations. Analytical Approaches Numerical Approaches Difference Methods Initial Value Methods 11. Special Topics. Reformulation of Problems in 'Standard' Form Generalized ODEs and Differential Algebraic Equations Eigenvalue Problems BVPs with Singularities Infinite Intervals Path Following, Singular Points and Bifurcation Highly Oscillatory Solutions Functional Differential Equations Method of Lines for PDEs Multipoint Problems On Code Design and Comparison Appendix A. A Multiple Shooting Code Appendix B. A Collocation Code References Bibliography Index.
1,154 citations