Showing papers on "Finite difference published in 2004"

The combined finite-discrete element method

Abstract: Preface.Acknowledgements.1 Introduction.1.1 General Formulation of Continuum Problems.1.2 General Formulation of Discontinuum Problems.1.3 A Typical Problem of Computational Mechanics of Discontinua.1.4 Combined Continua-Discontinua Problems.1.5 Transition from Continua to Discontinua.1.6 The Combined Finite-Discrete Element Method.1.7 Algorithmic and Computational Challenge of the Combined Finite-Discrete Element Method.2 Processing of Contact Interaction in the Combined Finite Discrete Element Method.2.1 Introduction.2.2 The Penalty Function Method.2.3 Potential Contact Force in 2D.2.4 Discretisation of Contact Force in 2D.2.5 Implementation Details for Discretised Contact Force in 2D.2.6 Potential Contact Force in 3D.2.6.1 Evaluation of contact force.2.6.2 Computational aspects.2.6.3 Physical interpretation of the penalty parameter.2.6.4 Contact damping.2.7 Alternative Implementation of the Potential Contact Force.3 Contact Detection.3.1 Introduction.3.2 Direct Checking Contact Detection Algorithm.3.2.1 Circular bounding box.3.2.2 Square bounding object.3.2.3 Complex bounding box.3.3 Formulation of Contact Detection Problem for Bodies of Similar Size in 2D.3.4 Binary Tree Based Contact Detection Algorithm for Discrete Elements of Similar Size.3.5 Direct Mapping Algorithm for Discrete Elements of Similar Size.3.6 Screening Contact Detection Algorithm for Discrete Elements of Similar Size.3.7 Sorting Contact Detection Algorithm for Discrete Elements of a Similar Size.3.8 Munjiza-NBS Contact Detection Algorithm in 2D.3.8.1 Space decomposition.3.8.2 Mapping of discrete elements onto cells.3.8.3 Mapping of discrete elements onto rows and columns of cells.3.8.4 Representation of mapping.3.9 Selection of Contact Detection Algorithm.3.10 Generalisation of Contact Detection Algorithms to 3D Space.3.10.1 Direct checking contact detection algorithm.3.10.2 Binary tree search.3.10.3 Screening contact detection algorithm.3.10.4 Direct mapping contact detection algorithm.3.11 Generalisation of Munjiza-NBS Contact Detection Algorithm to Multidimensional Space.3.12 Shape and Size Generalisation-Williams C-GRID Algorithm.4 Deformability of Discrete Elements.4.1 Deformation.4.2 Deformation Gradient.4.2.1 Frames of reference.4.2.2 Transformation matrices.4.3 Homogeneous Deformation.4.4 Strain.4.5 Stress.4.5.1 Cauchy stress tensor.4.5.2 First Piola-Kirchhoff stress tensor.4.5.3 Second Piola-Kirchhoff stress tensor.4.6 Constitutive Law.4.7 Constant Strain Triangle Finite Element.4.8 Constant Strain Tetrahedron Finite Element.4.9 Numerical Demonstration of Finite Rotation Elasticity in the Combined Finite-Discrete Element Method.5 Temporal Discretisation.5.1 The Central Difference Time Integration Scheme.5.1.1 Stability of the central difference time integration scheme.5.2 Dynamics of Irregular Discrete Elements Subject to Finite Rotations in 3D.5.2.1 Frames of reference.5.2.2 Kinematics of the discrete element in general motion.5.2.3 Spatial orientation of the discrete element.5.2.4 Transformation matrices.5.2.5 The inertia of the discrete element.5.2.6 Governing equation of motion.5.2.7 Change in spatial orientation during a single time step.5.6.8 Change in angular momentum due to external loads.5.6.9 Change in angular velocity during a single time step.5.6.10 Munjiza direct time integration scheme.5.3 Alternative Explicit Time Integration Schemes.5.3.1 The Central Difference time integration scheme (CD).5.3.2 Gear's predictor-corrector time integration schemes (PC-3, PC-4, and PC-5).5.3.3 CHIN integration scheme.5.3.4 OMF30 time integration scheme.5.3.5 OMF32 time integration scheme.5.3.6 Forest & Ruth time integration scheme.5.4 The Combined Finite-Discrete Element Simulation of the State of Rest.6 Sensitivity to Initial Conditions in Combined Finite-Discrete Element Simulations.6.1 Introduction.6.2 Combined Finite-Discrete Element Systems.7 Transition from Continua to Discontinua.7.1 Introduction.7.2 Strain Softening Based Smeared Fracture Model.7.3 Discrete Crack Model.7.4 A Need for More Robust Fracture Solutions.8 Fluid Coupling in the Combined Finite-Discrete Element Method.8.1 Introduction.8.1.1 CFD with solid coupling.8.1.2 Combined finite-discrete element method with CFD coupling.8.2 Expansion of the Detonation Gas.8.2.1 Equation of state.8.2.2 Rigid chamber.8.2.3 Isentropic adiabatic expansion of detonation gas.8.2.4 Detonation gas expansion in a partially filled non-rigid chamber.8.3 Gas Flow Through Fracturing Solid.8.3.1 Constant area duct.8.4 Coupled Combined Finite-Discrete Element Simulation of Explosive Induced Fracture and Fragmentation.8.4.1 Scaling of coupled combined finite-discrete element problems.8.5 Other Applications.9 Computational Aspects of Combined Finite-Discrete Element Simulations.9.1 Large Scale Combined Finite-Discrete Element Simulations.9.1.1 Minimising RAM requirements.9.1.2 Minimising CPU requirements.9.1.3 Minimising storage requirements.9.1.4 Minimising risk.9.1.5 Maximising transparency.9.2 Very Large Scale Combined Finite-Discrete Element Simulations.9.3 Grand Challenge Combined Finite-Discrete Element Simulations.9.4 Why the C Programming Language?9.5 Alternative Hardware Architectures.9.5.1 Parallel computing.9.5.2 Distributed computing.9.5.3 Grid computing.10 Implementation of some of the Core Combined Finite-Discrete Element Algorithms.10.1 Portability, Speed, Transparency and Reusability.10.1.1 Use of new data types.10.1.2 Use of MACROS.10.2 Dynamic Memory Allocation.10.3 Data Compression.10.4 Potential Contact Force in 3D.10.4.1 Interaction between two tetrahedrons.10.5 Sorting Contact Detection Algorithm.10.6 NBS Contact Detection Algorithm in 3D.10.7 Deformability with Finite Rotations in 3D.Bibliography.Index.

Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: and Well-Balanced Schemes for Sources

Abstract: Introduction.- 1. Quasilinear systems and conservation laws.- 2. Conservative schemes.- 3. Source terms.- 4. Nonconservative schemes.- 5. Multidimensional finite volumes with sources.- 6. Numerical test with source.- Bibliography

Computing the Ground State Solution of Bose--Einstein Condensates by a Normalized Gradient Flow

Abstract: In this paper, we present a continuous normalized gradient flow (CNGF) and prove its energy diminishing property, which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of Bose--Einstein condensates (BEC). We also investigate the energy diminishing property for the discretization of the CNGF. Two numerical methods are proposed for such discretizations: one is the backward Euler centered finite difference (BEFD) method, the other is an explicit time-splitting sine-spectral (TSSP) method. Energy diminishing for BEFD and TSSP for the linear case and monotonicity for BEFD for both linear and nonlinear cases are proven. Comparison between the two methods and existing methods, e.g., Crank--Nicolson finite difference (CNFD) or forward Euler finite difference (FEFD), shows that BEFD and TSSP are much better in terms of preserving the energy diminishing property of the CNGF. Numerical results in one, two, and three dimensions with magnetic trap confinement potential, as well as a potential of a stirrer corresponding to a far-blue detuned Gaussian laser beam, are reported to demonstrate the effectiveness of BEFD and TSSP methods. Furthermore we observe that the CNGF and its BEFD discretization can also be applied directly to compute the first excited state solution in BEC when the initial data is chosen as an odd function.

A Fast Apparent‐Horizon Finder for 3‐Dimensional Cartesian Grids in Numerical Relativity

Abstract: In 3 + 1 numerical simulations of dynamic black-hole spacetimes, it is useful to be able to find the apparent horizon(s) (AH) in each slice of a time evolution. A number of AH finders are available, but they often take many minutes to run, so they are too slow to be practically usable at each time step. Here I present a new AH finder, AHFINDERDIRECT, which is very fast and accurate: at typical resolutions it takes only a few seconds to find an AH to ~10−5m accuracy on a GHz-class processor. I assume that an AH to be searched for is a Strahlkorper ('star-shaped region') with respect to some local origin, and so parametrize the AH shape by r = h(angle) for some single-valued function h:S2 → +. The AH equation then becomes a nonlinear elliptic PDE in h on S2, whose coefficients are algebraic functions of gij, Kij, and the Cartesian-coordinate spatial derivatives of gij. I discretize S2 using six angular patches (one each in the neighbourhood of the ±x, ± y, and ±z axes) to avoid coordinate singularities, and finite difference the AH equation in the angular coordinates using fourth-order finite differencing. I solve the resulting system of nonlinear algebraic equations (for h at the angular grid points) by Newton's method, using a 'symbolic differentiation' technique to compute the Jacobian matrix. AHFINDERDIRECT is implemented as a thorn in the CACTUS computational toolkit, and is freely available by anonymous CVS checkout.

High resolution algorithms for multidimensional population balance equations

Abstract: Population balance equations have been used to model a wide range of processes including polymerization, crystallization, cloud formation, and cell dynamics. Rather than developing new algorithms specific to population balance equations, it is proposed to adapt the high-resolution finite volume methods developed for compressible gas dynamics, which have been applied to aerodynamics, astrophysics, detonation waves, and related fields where shock waves occur. High-resolution algorithms are presented for simulating multidimensional population balance equations with nucleation and size-dependent growth rates. For sharp distributions, these high-resolution algorithms can achieve improved numerical accuracy with orders-of-magnitude lower computational cost than other finite difference and finite volume algorithms. The algorithms are implemented in the ParticleSolver software package, which is applied to batch and continuous processes with one and multiple internal coordinates. © 2004 American Institute of Chemical Engineers AIChE J, 50: 2738 –2749, 2004

Noise removal using smoothed normals and surface fitting

Abstract: In this work, we use partial differential equation techniques to remove noise from digital images. The removal is done in two steps. We first use a total-variation filter to smooth the normal vectors of the level curves of a noise image. After this, we try to find a surface to fit the smoothed normal vectors. For each of these two stages, the problem is reduced to a nonlinear partial differential equation. Finite difference schemes are used to solve these equations. A broad range of numerical examples are given in the paper.

Solving the three-body Coulomb breakup problem using exterior complex scaling

Abstract: Electron-impact ionization of the hydrogen atom is the prototypical three-body Coulomb breakup problem in quantum mechanics. The combination of subtle correlation effects and the difficult boundary conditions required to describe two electrons in the continuum have made this one of the outstanding challenges of atomic physics. A complete solution of this problem in the form of a 'reduction to computation' of all aspects of the physics is given by the application of exterior complex scaling, a modern variant of the mathematical tool of analytic continuation of the electronic coordinates into the complex plane that was used historically to establish the formal analytic properties of the scattering matrix. This review first discusses the essential difficulties of the three-body Coulomb breakup problem in quantum mechanics. It then describes the formal basis of exterior complex scaling of electronic coordinates as well as the details of its numerical implementation using a variety of methods including finite difference, finite elements, discrete variable representations and B-splines. Given these numerical implementations of exterior complex scaling, the scattering wavefunction can be generated with arbitrary accuracy on any finite volume in the space of electronic coordinates, but there remains the fundamental problem of extracting the breakup amplitudes from it. Methods are described for evaluating these amplitudes. The question of the volume-dependent overall phase that appears in the formal theory of ionization is resolved. A summary is presented of accurate results that have been obtained for the case of electron-impact ionization of hydrogen as well as a discussion of applications to the double photoionization of helium.

On Quadrature Methods for Highly Oscillatory Integrals and Their Implementation

Abstract: The main theme of this paper is the construction of efficient, reliable and affordable error bounds for two families of quadrature methods for highly oscillatory integrals. We demonstrate, using asymptotic expansions, that the error can be bounded very precisely indeed at the cost of few extra derivative evaluations. Moreover, in place of derivatives it is possible to use finite difference approximations, with spacing inversely proportional to frequency. This renders the computation of error bounds even cheaper and, more importantly, leads to a new family of quadrature methods for highly oscillatory integrals that can attain arbitrarily high asymptotic order without computation of derivatives.

Stable and Accurate Artificial Dissipation

Abstract: Stability for nonlinear convection problems using centered difference schemes require the addition of artificial dissipation. In this paper we present dissipation operators that preserve both stability and accuracy for high order finite difference approximations of initial boundary value problems.

New Difference Schemes for Partial Differential Equations

Abstract: 1 Linear Difference Equations.- 1.1 Difference Equations of the First Order.- 1.2 Difference Equations of the Second Order.- 1.3 Difference Equations with Constant Coefficients.- 2 Difference Schemes for First-Order Differential Equations.- 2.1 Single-Step Exact Difference Scheme and Its Applications.- 2.2 Taylor's Decomposition on Two Points and Its Applications.- 3 Difference Schemes for Second-Order Differential Equations.- 3.1 Two-Step Exact Difference Scheme and Its Applications.- 3.2 Taylor's Decomposition on Three Points and Its Applications.- 4 Partial Differential Equations of Parabolic Type.- 4.1 A Cauchy Problem. Well-posedness.- 4.2 Difference Schemes Generated by an Exact Difference Scheme.- 4.3 Single-Step Difference Schemes Generated by Taylor's Decomposition.- 5 Partial Differential Equations of Elliptic Type.- 5.1 A Boundary-Value Problem. Well-posedness.- 5.2 Difference Schemes Generated by an Exact Difference Scheme.- 5.3 Two-Step Difference Schemes Generated by Taylor's Decomposition.- 6 Partial Differential Equations of Hyperbolic Type.- 6.1 A Cauchy Problem.- 6.2 Difference Schemes Generated by an Exact Difference Scheme.- 6.3 Two-Step Difference Schemes Generated by Taylor's Decomposition.- 7 Uniform Difference Schemes for Perturbation Problems.- 7.1 A Cauchy Problem for Parabolic Equations.- 7.2 A Boundary-Value Problem for Elliptic Equations.- 7.3 A Cauchy Problem for Hyperbolic Equations.- 8 Appendix: Delay Parabolic Differential Equations.- 8.1 The Initial-Value Differential Problem.- 8.2 The Difference Schemes.- Comments on the Literature.

A parallel finite-difference approach for 3D transient electromagnetic modeling with galvanic sources

Abstract: A parallel finite-difference algorithm for the solution of diffusive, three-dimensional (3D) transient electromagnetic field simulations is presented. The purpose of the scheme is the simulation of both electric fields and the time derivative of magnetic fields generated by galvanic sources (grounded wires) over arbitrarily complicated distributions of conductivity and magnetic permeability. Using a staggered grid and a modified DuFort-Frankel method, the scheme steps Maxwell's equations in time. Electric field initialization is done by a conjugate-gradient solution of a 3D Poisson problem, as is common in 3D resistivity modeling. Instead of calculating the initial magnetic field directly, its time derivative and curl are employed in order to advance the electric field in time. A divergence-free condition is enforced for both the magnetic-field time derivative and the total conduction-current density, providing accurate results at late times. In order to simulate large realistic earth models, the algorithm has been designed to run on parallel computer platforms. The upward continuation boundary condition for a stable solution in the infinitely resistive air layer involves a two-dimensional parallel fast Fourier transform. Example simulations are compared with analytical, integral-equation and spectral Lanczos decomposition solutions and demonstrate the accuracy of the scheme.

Wave and Scattering Methods for Numerical Simulation

Abstract: Preface.Foreword.1. Introduction.1.1 An Overview of Scattering Methods.1.1.1 Remarks on Passivity.1.1.2 Case Study: The Kelly-Lochbaum Digital Speech Synthesis Mode.1.1.3 Digital Waveguide Networks.1.1.4 A General Approach: Multidimensional Circuit Representations and Wave Digital Filters.1.2 Questions.2. Wave Digital Filters.2.1 Classical Network Theory.2.1.1 N-ports.2.1.2 Power and Passivity.2.1.3 Kirchhoff's Laws.2.1.4 Circuit Elements.2.2 Wave Digital Elements and Connections.2.2.1 The Bilinear Transform.2.2.2 Wave Variables.2.2.3 Pseudopower and Pseudopassivity.2.2.4 Wave Digital Elements.2.2.5 Adaptors.2.2.6 Signal and Coefficient Quantization.2.2.7 VectorWave Variables.2.3 Wave Digital Filters and Finite Differences.3. Multidimensional Wave Digital Filters.3.1 Symmetric Hyperbolic Systems.3.2 Coordinate Changes and Grid Generation.3.2.1 Structure of Coordinate Changes.3.2.2 Coordinate Changes in (1 +1)D.3.2.3 Coordinate Changes in Higher Dimensions.3.3 MD-passivity.3.4 MD Circuit Elements.3.4.1 The MD Inductor.3.4.2 OtherMD Elements.3.4.3 Discretization in the Spectral Domain.3.4.4 Other Spectral Mappings.3.5 The (1 +1)D Advection Equation.3.5.1 A Multidimensional Kirchhoff Circuit.3.5.2 Stability.3.5.3 An Upwind Form.3.6 The (1 +1)D Transmission Line.3.6.1 MDKC for the (1 + 1)D Transmission Line Equations.3.6.2 Digression: The Inductive Lattice Two-port.3.6.3 Energetic Interpretation.3.6.4 A MDWD Network for the (1 + 1)D Transmission Line.3.6.5 Simplified Networks.3.7 The (2 +1)D Parallel-plate System.3.7.1 MDKC and MDWD Network.3.8 Finite-difference Interpretation.3.8.1 MDWD Networks as Multistep Schemes.3.8.2 Numerical Phase Velocity and Parasitic Modes.3.9 Initial Conditions.3.10 Boundary Conditions.3.10.1 MDKC Modeling of Boundaries.3.11 Balanced Forms.3.12 Higher-order Accuracy.4. Digital Waveguide Networks.4.1 FDTD and TLM.4.2 Digital Waveguides.4.2.1 The Bidirectional Delay Line.4.2.2 Impedance.4.2.3 Wave Equation Interpretation.4.2.4 Note on the Different Definitions of Wave Quantities.4.2.5 Scattering Junctions.4.2.6 Vector Waveguides and Scattering Junctions.4.2.7 Transitional Note.4.3 The (1 +1)D Transmission Line.4.3.1 First-order System and the Wave Equation .4.3.2 Centered Difference Schemes and Grid Decimation.4.3.3 A (1+1)D Waveguide Network.4.3.4 Waveguide Network and the Wave Equation.4.3.5 An Interleaved Waveguide Network.4.3.6 Varying Coefficients.4.3.7 Incorporating Losses and Sources.4.3.8 Numerical Phase Velocity and Dispersion.4.3.9 Boundary Conditions.4.4 The (2 +1)D Parallel-plate System.4.4.1 Defining Equations and Centered Differences.4.4.2 The Waveguide Mesh.4.4.3 Reduced Computational Complexity and Memory Requirements in the Standard Form of the Waveguide Mesh.4.4.4 Boundary Conditions.4.5 Initial Conditions.4.6 Music and Audio Applications of Digital Waveguides.5. Extensions of Digital Waveguide Networks.5.1 Alternative Grids in (2 +1)D.5.1.1 Hexagonal and Triangular Grids.5.1.2 The Waveguide Mesh in Radial Coordinates.5.2 The (3 + 1)D Wave Equation and Waveguide Meshes.5.3 The Waveguide Mesh in General Curvilinear Coordinates.5.4 Interfaces between Grids.5.4.1 Doubled Grid Density Across an Interface.5.4.2 Progressive Grid Density Doubling.5.4.3 Grid Density Quadrupling.5.4.4 Connecting Rectilinear and Radial Grids.5.4.5 Grid Density Doubling in (3 +1)D.5.4.6 Note.6. Incorporating the DWN into the MDWD Framework.6.1 The (1 +1)D Transmission Line Revisited.6.1.1 Multidimensional Unit Elements.6.1.2 Hybrid Form of the Multidimensional Unit Element.6.1.3 Alternative MDKC for the (1+1)D Transmission Line.6.2 Alternative MDKC for the (2 + 1)D Parallel-plate System.6.3 Higher-order Accuracy Revisited.6.4 Maxwell's Equations.7. Applications to Vibrating Systems.7.1 Beam Dynamics.7.1.1 MDKC and MDWDF for Timoshenko's System.7.1.2 Waveguide Network for Timoshenko's System.7.1.3 Boundary Conditions in the DWN.7.1.4 Simulation: Timoshenko's System for Beams of Uniform and Varying Cross-sectional Areas.7.1.5 Improved MDKC for Timoshenko's System via Balancing.7.2 Plates.7.2.1 MDKCs and Scattering Networks for Mindlin's System.7.2.2 Boundary Termination of the Mindlin Plate.7.2.3 Simulation: Mindlin's System for Plates of Uniform and Varying Thickness.7.3 Cylindrical Shells.7.3.1 The Membrane Shell.7.3.2 The Naghdi-Cooper System II Formulation.7.4 Elastic Solids.7.4.1 Scattering Networks for the Navier System.7.4.2 Boundary Conditions.8. Time-varying and Nonlinear Systems.8.1 Time-varying and Nonlinear Circuit Elements.8.1.1 Lumped Elements.8.1.2 Distributed Elements.8.2 Linear Time-varying Distributed Systems.8.2.1 A Time-varying Transmission Line Model.8.3 Lumped Nonlinear Systems in Musical Acoustics.8.3.1 Piano Hammers.8.3.2 The Single Reed.8.4 From Wave Digital Principles to Relativity Theory.8.4.1 Origin of the Challenge.8.4.2 The Principle of Newtonian Limit.8.4.3 Newton's Second Law.8.4.4 Newton's Third Law and Some Consequences.8.4.5 Moving Electromagnetic Field.8.4.6 The Bertozzi Experiment.8.5 Burger's Equation.8.6 The Gas Dynamics Equations.8.6.1 MDKC and MDWDF for the Gas Dynamics Equations.8.6.2 An Alternate MDKC and Scattering Network.8.6.3 Entropy Variables.9. Concluding Remarks.9.1 Answers.9.2 Questions.A. Finite Difference Schemes for the Wave Equation.A.1 Von Neumann Analysis of Difference Schemes.A.1.1 One-step Schemes.A.1.2 Multistep Schemes.A.1.3 Vector Schemes.A.1.4 Numerical Phase Velocity.A.2 Finite Difference Schemes for the (2 + 1)D Wave Equation.A.2.1 The Rectilinear Scheme.A.2.2 The Interpolated Rectilinear Scheme.A.2.3 The Triangular Scheme.A.2.4 The Hexagonal Scheme.A.2.5 Note on Higher-order Accuracy.A.3 Finite Difference Schemes for the (3 + 1)D Wave Equation.A.3.1 The Cubic Rectilinear Scheme.A.3.2 The Octahedral Scheme.A.3.3 The (3 + 1)D Interpolated Rectilinear Scheme.A.3.4 The Tetrahedral Scheme.B. Eigenvalue and Steady State Problems.B.1 Introduction.B.2 Abstract Time Domain Models.B.3 Typical Eigenvalue Distribution of a Discretized PDE.B.4 Excitation and Filtering.B.5 Partial Similarity Transform.B.6 Steady State Problems.B.7 Generalization to Multiple Eigenvalues.B.8 Numerical Example.Bibliography.Index.

Applied partial differential equations

Abstract: 1. Heat Equation. 2. Method of Separation of Variables. 3. Fourier Series. 4. Vibrating Strings and Membranes. 5. Sturm-Liouville Eigenvalue Problems. 6. Finite Difference Numerical Methods for Partial Differential Equations. 7. Partial Differential Equations with at Least Three Independent Variables. 8. Nonhomogeneous Problems. 9. Green's Functions for Time-Independent Problems. 10. Infinite Domain Problems-Fourier Transform Solutions of Partial Differential Equations. 11. Green's Functions for Wave and Heat Equations. 12. The Method of Characteristics for Linear and Quasi-Linear Wave Equations. 13. A Brief Introduction to Laplace Transform Solution of Partial Differential Equations. 14. Topics: Dispersive Waves, Stability, Nonlinearity, and Perturbation Methods. Bibliography. Selected Answers to Starred Exercises. Index.

Godunov-Type Methods for Conservation Laws with a Flux Function Discontinuous in Space

Abstract: Scalar conservation laws with a flux function discontinuous in space are approximated using a Godunov-type method for which a convergence theorem is proved. The case where the flux functions at the interface intersect is emphasized. A very simple formula is given for the interface flux. A numerical comparison between the Godunov numerical flux and the upstream mobility flux is presented for two-phase flow in porous media. A consequence of the convergence theorem is an existence theorem for the solution of the scalar conservation laws under consideration. Furthermore, for regular solutions, uniqueness has been shown.

Sounding of finite solid bodies by way of topological derivative

Abstract: This paper is concerned with an application of the concept of topological derivative to elastic-wave imaging of finite solid bodies containing cavities. Building on the approach originally proposed in the (elastostatic) theory of shape optimization, the topological derivative, which quantifies the sensitivity of a featured cost functional due to the creation of an infinitesimal hole in the cavity-free (reference) body, is used as a void indicator through an assembly of sampling points where it attains negative values. The computation of topological derivative is shown to involve an elastodynamic solution to a set of supplementary boundary-value problems for the reference body, which are here formulated as boundary integral equations. For a comprehensive treatment of the subject, formulas for topological sensitivity are obtained using three alternative methodologies, namely (i) direct differentiation approach, (ii) adjoint field method, and (iii) limiting form of the shape sensitivity analysis. The competing techniques are further shown to lead to distinct computational procedures. Methodologies (i) and (ii) are implemented within a BEM-based platform and validated against an analytical solution. A set of numerical results is included to illustrate the utility of topological derivative for 3D elastic-wave sounding of solid bodies; an approach that may perform best when used as a pre-conditioning tool for more accurate, gradient-based imaging algorithms. Despite the fact that the formulation and results presented in this investigation are established on the basis of a boundary integral solution, the proposed methodology is readily applicable to other computational platforms such as the finite element and finite difference techniques.

Possibilities of finite calculus in computational mechanics

Abstract: The expression “finite calculus” refers to the derivation of the governing differential equations in mechanics by invoking balance of fluxes, forces, etc. in a domain of finite size. The governing equations resulting from this approach are different from those of infinitessimal calculus theory and they incorporate new terms depending on the dimensions of the balance domain. The new modified equations allow to derive naturally stabilized numerical schemes using finite element, finite difference, finite volume or meshless methods. The paper briefly discusses the possibilities of the modified governing equations derived via the finite calculus technique for the numerical solution of convection-diffusion problems, incompressible flow and incompressible solid mechanic problems and strain localization problems.

Nonlinearity in atmospheric response: A direct sensitivity analysis approach

Abstract: [1] The decoupled direct method (DDM) is used for efficient and accurate calculation of the higher-order sensitivity coefficients in a regional photochemical air quality model with detailed chemical mechanism (Statewide Air Pollution Research Center (SAPRC-99)). High-order DDM (HDDM) is an extension to a previous implementation of DDM in three-dimensional air quality models (DDM-3D) that directly calculates the higher-order derivatives (with respect to one parameter, as well as cross derivatives) with similar computational efficiency as the first-order implementation and is also modified for better accuracy. (H)DDM results show very good agreement with brute force (finite difference) sensitivity coefficients for the first- and second-order derivatives, but the agreement deteriorates for higher-order coefficients. The nature of the truncation errors and other inaccuracies in the brute force approximations are explored. The difference between the first-order brute force and DDM derivatives is dominated (and largely explained) by the truncation errors as calculated from HDDM results. Taylor expansion is used for parametric scaling of the response with the use of sensitivity coefficients. Use of higher-order coefficients can significantly improve the accuracy of such projections. Finally, higher-order sensitivity coefficients of ozone with respect to NOx and volatile organic compound emissions (including cross derivatives) are used to create time- and location-dependent ozone isopleths.