Solution of Index 2 Implicit Differential-Algebraic Equations by Lobatto Runge-Kutta Methods

We consider the numerical solution of systems of index 2 implicit differential-algebraic equations (DAEs) by a class of super partitioned additive Runge–Kutta (SPARK) methods. The families of Lobatto IIIA-B-C-C^*-D methods are included. We show super-convergence of optimal order 2s–2 for the s-stage Lobatto families provided the constraints are treated in a particular way which strongly relies on specific properties of the SPARK coefficients. Moreover, reversibility properties of the flow can still be preserved provided certain SPARK coefficients are symmetric.     

Modelling dispersion in laminar and turbulent flows in an open channel based on centre manifolds using 1D-IRBFN method

Centre manifold method is an accurate approach for analytically constructing an advection–diffusion equation (and even more accurate equations involving higher-order derivatives) for the depth-averaged concentration of substances in channels. This paper presents a direct numerical verification of this method with examples of the dispersion in laminar and turbulent flows in an open channel with a smooth bottom. The one-dimensional integrated radial basis function network (1D-IRBFN) method is used as a numerical approach to obtain a numerical solution for the original two-dimensional (2-D) advection–diffusion equation. The 2-D solution is depth-averaged and compared with the solution of the 1-D equation derived using the centre manifolds. The numerical results show that the 2-D and 1-D solutions are in good agreement both for the laminar flow and turbulent flow. The maximum depth-averaged concentrations for the 1-D and 2-D models gradually converge to each other, with their velocities becoming practically equal. The obtained numerical results also demonstrate that the longitudinal diffusion can be neglected compared to the advection.     

Multi-scale B-spline method for 2-D elastic problems

In this paper, quadratic B-spline functions are used for solution of 2-D elastic problems. Because B-spline functions are directly used as basis function, there is no need to use meshes and nodes in function approximation. In order to improve the computational efficiency, different scales are used for sub-domains of entire problem domain in function approximation. The modified variational form and Lagrange multipliers method are used for coupling of different scale in function approximation. Compared with meshless methods and other wavelet based methods, this multi-scale B-spline-based method is simple and easy to work with for numerical analysis. Furthermore, the computational efficiency of the multi-scale method is much higher than that of single scale approach. The numerical examples of 2-D elastic problems indicate that the present method is effective and stable for solving complicated problems.     

A new stabilizing technique for boundary integral methods for water waves

Boundary integral methods to compute interfacial flows are very sensitive to numerical instabilities. A previous stability analysis by Beale, Hou and Lowengrub reveals that a very delicate balance among terms with singular integrals and derivatives must be preserved at the discrete level in order to maintain numerical stability. Such balance can be preserved by applying suitable numerical filtering at certain places of the discretization. While this filtering technique is effective for two-dimensional (2-D) periodic fluid interfaces, it does not apply to nonperiodic fluid interfaces. Moreover, using the filtering technique alone does not seem to be sufficient to stabilize 3-D fluid interfaces.

Here we introduce a new stabilizing technique for boundary integral methods for water waves which applies to nonperiodic and 3-D interfaces. A stabilizing term is added to the boundary integral method which exactly cancels the destabilizing term produced by the point vortex method approximation to the leading order. This modified boundary integral method still has the same order of accuracy as the point vortex method. A detailed stability analysis is presented for the point vortex method for 2-D water waves. The effect of various stabilizing terms is illustrated through careful numerical experiments.

     

Numerical solution of the incompressible Navier-Stokes equations by Krylov subspace and multigrid methods

We consider numerical solution methods for the incompressible Navier-Stokes equations discretized by a finite volume method on staggered grids in general coordinates. We use Krylov subspace and multigrid methods as well as their combinations. Numerical experiments are carried out on a scalar and a vector computer. Robustness and efficiency of these methods are studied. It appears that good methods result from suitable combinations of GCR and multigrid methods.     

H-p clouds—an h-p meshless method

A new methodology to build discrete models of boundary-value problems is presented. The h-p cloud method is applicable to arbitrary domains and employs only a scattered set of nodes to build approximate solutions to BVPs. This new method uses radial basis functions of varying size of supports and with polynomial-reproducing properties of arbitrary order. The approximating properties of the h-p cloud functions are investigated in this article and several theorems concerning these properties are presented. Moving least squares interpolants are used to build a partition of unity on the domain of interest. These functions are then used to construct, at a very low cost, trial and test functions for Galerkin approximations. The method exhibits a very high rate of convergence and has a greater flexibility than traditional h-p finite element methods. Several numerical experiments in 1-D and 2-D are also presented. © 1996 John Wiley & Sons, Inc.     

A consistent theory for poroelastic plates

In many fields of engineering thin porous components are used, e.g. as damping elements for noise insulation in cars or walls in buildings. Today these elements are often calculated using a numerical 3-D model. Because of numerical problems which occur using a 3-D model for thin transversly loaded structures a plate theory is advantageous. To take into account the porous structure as well as the damping effect of the porosity of these components a poroelastic plate theory is necessary. Several posibilities exist to establish plate theories. Generally, methods to derive a plate theory require a priory assumptions motivated by engineering intuition (like the classical Kirchhoff normal hypothesis). In this contribution a priori assumptions are not used. Plate theories of different orders are derived from the 3-D poroelastic theory using series expansion. For elastic plates this idea was introduced in [3]. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Digital Rock Physics: A case study of carbonate rocks

Digital Rock physics is a numerical framework applying image-based methods to determine effective material properties of rock. It is possible to find several of these methods in the literature, but we focused mainly in micro X-Ray Computed Tomography (µXRCT) to acquire the digital rock samples. After scanning the 3-D microstructure, morphological filters and numerical simulations were performed to characterize the samples. In particular carbonate rocks present a mismatch in numerical results compared to laboratory experiments. The elastic parameters, the P- and S-wave moduli, respectively,are overestimated when compared to ultrasonic measurements in the laboratory. We describe possible causes of this mismatch and propose a new segmentation technique to improve the correlation between numerical simulations and laboratory data. Furthermore, the workflow proposed for the characterization of carbonate rocks has been applied to different digitized samples from different sources, and a data driven material law has been found that fits better experimental results than the lower Hashin-Shtrikman bound. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A geometric view of Krylov subspace methods on singular systems

We give a geometric framework for analysing iterative methods on singular linear systems A x = b and apply them to Krylov subspace methods. The idea is to decompose the method into the ?(A) component and its orthogonal complement ?(A)^?, where ?(A) is the range of A. We apply the framework to GMRES, GMRES(k) and GCR(k), and derive conditions for convergence without breakdown for inconsistent and consistent singular systems. The approach also gives a geometric interpretation and different proofs of the conditions obtained by Brown and Walker for GMRES. We also give examples arising in the finite difference discretization of two‐point boundary value problems of an ordinary differential equation. Copyright © 2010 John Wiley & Sons, Ltd.     

Structural Optimization with FEM-based Shakedown Analyses

In this paper, a mathematical programming formulation is presented for the structural optimization with respect to the shakedown analysis of 3-D perfectly plastic structures on basis of a finite element discretization. A new direct algorithm using plastic sensitivities is employed in solving this optimization formulation. The numerical procedure has been applied to carry out the shakedown analysis of pipe junctions under multi-loading systems. The new approach is compared to so-called derivative-free direct search methods. The computational effort of the proposed method is much lower compared to this methods.     

A quadratic b-spline based isogeometric analysis of transient wave propagation problems with implicit time integration method

A uniform quadratic b-spline isogeometric element is exclusively considered for wave propagation problem with the use of desirable implicit time integration scheme. A generalized numerical algorithm is proposed for dispersion analysis of one-dimensional (1-D) and two-dimensional (2-D) wave propagation problems where the quantified influence of the defined CFL number on wave velocity error is analyzed and obtained. Meanwhile, the optimal CFL (Courant–Friedrichs–Lewy) number for the proposed 1-D and 2-D problems is suggested. Four representative numerical simulations confirm the effectiveness of the proposed method and the correctness of dispersion analysis when appropriate spatial element size and time increment are adopted. The desirable computation efficiency of the proposed isogeometric method was confirmed by conducting time cost and calculation accuracy analysis of a 2-D numerical example where the referred FEM was also tested for comparison.     

Experiments and Identifications for Finite Polymer Inelasticity

The constitutive theories intended to quantitatively account for the complicated material response exhibited by polymers include, in general, adjustable material parameters. These must be identified from experimental data obtained from the material under consideration. This contribution presents the complete procedure studying the behavior of polymers at large strains in three basic steps: i) Accomplishment of homogeneous and 3-D inhomogeneous experiments under different deformation conditions. ii) Identification of the material parameters of a constitutive model by means of gradient–based optimization methods with respect to the homogeneous experimental data. iii) Validation of the identified material parameters by comparing 3-D FE simulations to the inhomogeneous experimental data. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method

In this article one discusses the controllability of a semi-discrete system obtained by discretizing in space the linear 1-D wave equation with a boundary control at one extremity. It is known that the semi-discrete models obtained with finite difference or the classical finite element method are not uniformly controllable as the discretization parameter h goes to zero (see [8]). Here we introduce a new semi-discrete model based on a mixed finite element method with two different basis functions for the position and velocity. We show that the controls obtained with these semi-discrete systems can be chosen uniformly bounded in L²(0,T) and in such a way that they converge to the HUM control of the continuous wave equation, i.e. the minimal L²-norm control. We illustrate the mathematical results with several numerical experiments. Supported by Grant BFM 2002-03345 of MCYT (Spain) and the TMR projects of the EU ``Homogenization and Multiple Scales" and ``New materials, adaptive systems and their nonlinearities: modelling, control and numerical simulations". Partially Supported by Grant BFM 2002-03345 of MCYT (Spain), Grant 17 of Egide-Brancusi Program and Grant 80/2005 of CNCSIS (Romania).     

Variable stepsize variable formula methods based on predictor-corrector schemes

The use of fairly general predictor-corrector (PC) schemes of linear multistep (LM) formulae in the numerical solution of systems of ordinary differential equations (ODE's) is considered. It is assumed that both the stepsize and the PC scheme can be varied during the computational process. The numerical methods obtained under these two assumptions are called predictor-corrector linear multistep variable stepsize variable formula methods (PC LM VSVFM's). The consistency, zero-stability and convergence properties of the PC LM VSVFM's are studied. Several results concerning these fundamental properties of the numerical methods are established. It should be emphasized that all theorems are formulated and proved under very mild assumptions on the stepsize selection strategy. The extension of the results for the so-called one-leg methods is briefly discussed. The use of PC LM VSVFM's leads to a very efficient treatment of many mathematical models describing different phenomena in science and engineering. Such methods have successfully been used in the numerical solution of systems of ODE's arising after the space discretization of some air pollution models.     

Remarks on the numerical treatment of Prandtl's boundary-layer equations

Summary The Crocco Transformation for the boundary-layer equations very successfully used by Oleinik and Nickel for theoretical purposes (existence of 2-D, uniqueness of 2-D and 3-D solutions) will be used here for the construction of numerical procedures. An approximate discretization of the transformed equations leads to a nicely structured system of nonlinear equations that can be devided into small parts to be solved one after the other (in the 3-D case even partially by parallel processing). The Jacobians of these parts are oftenM-matrices such that SOR iterations work. The inverse transformation can numerically be realized very simply.

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Zusammenfassung Die für theoretische Zwecke (Existenz von Lösungen im 2-D-Fall, Eindeutigkeit im 2-D- und 3-D-Fall) von Oleinik und Nickel so erfolgreich eingesetzte Crocco-Transformation der Grenzschichtgleichungen wird in der vorliegenden Arbeit zur Konstruktion numerischer Verfahren benutzt. Bei geeigneter Diskretisierung erhält man angenehm strukturierte Systeme nichtlinearer Gleichungen, die in sukzessiv aufzurufende und unabhängig zu lösende Teil-Systeme zerfallen (im 3-D-Fall können die Systeme teilweise parallel bearbeitet werden). Die Jacobi-Matrizen der Teil-Systeme sind oftmalsM-Matrizen, so daß SOR-Iterationsverfahren konvergieren. Die Rücktransformation erfordert numerisch nur geringen Aufwand.

     

The convergence of Krylov subspace methods for large unsymmetric linear systems

The convergence problem of many Krylov subspace methods,e.g., FOM, GCR, GMRES and QMR, for solving large unsymmetric (non-Hermitian) linear systems is considered in a unified way when the coefficient matrixA is defective and its spectrum lies in the open right (left) half plane. Related theoretical error bounds are established and some intrinsic relationships between the convergence speed and the spectrum ofA are exposed. It is shown that these methods are likely to converge slowly once one of the three cases occurs:A is defective, the distribution of its spectrum is not favorable, or the Jordan basis ofA is ill conditioned. In the proof, some properties on the higher order derivatives of Chebyshev polynomials in an ellipse in the complex plane are derived, one of which corrects a result that has been used extensively in the literature. Supported by the China State Major Key Project for Basic Researches, the National Natural Science Foundation of China, the Doctoral Program of the Chinese National Educational Commission, the Foundation of Returned Scholars of China and Liaoning Province Natural Science Foundation.     

A Conservative Space-time Mesh Refinement Method for the 1-D Wave Equation. Part II: Analysis

Summary. We introduced in [2] a new method for space-time refinement for the 1-D wave equation. This method is based on the conservation of a discrete energy through two different discretization grids which guarantees the stability of the scheme. In this second part, we analyse the accuracy of this scheme in a detailed way by means of a plane wave analysis and numerical experiments that permit us to point out spurious numerical phenomena and explain how to control them. Mathematics Subject Classification (2000):65N12     

High orderP-stable formulae for the numerical integration of periodic initial value problems

Summary Recently there has been considerable interest in the approximate numerical integration of the special initial value problemy=f(x, y) for cases where it is known in advance that the required solution is periodic. The well known class of Störmer-Cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems. An appropriate stability requirement for the numerical integration of periodic problems is that ofP-stability. However Lambert and Watson have shown that aP-stable linear multistep method cannot have an order of accuracy greater than 2. In the present paper a class of 2-step methods of Runge-Kutta type is discussed for the numerical solution of periodic initial value problems.P-stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods.     

BDDC methods for discontinuous Galerkin discretization of elliptic problems

A discontinuous Galerkin (DG) discretization of Dirichlet problem for second-order elliptic equations with discontinuous coefficients in 2-D is considered. For this discretization, balancing domain decomposition with constraints (BDDC) algorithms are designed and analyzed as an additive Schwarz method (ASM). The coarse and local problems are defined using special partitions of unity and edge constraints. Under certain assumptions on the coefficients and the mesh sizes across ∂Ω_i, where the Ω_i are disjoint subregions of the original region Ω, a condition number estimate C(1+max_ilog(H_i/h_i))² is established with C independent of h_i, H_i and the jumps of the coefficients. The algorithms are well suited for parallel computations and can be straightforwardly extended to the 3-D problems. Results of numerical tests are included which confirm the theoretical results and the necessity of the imposed assumptions.     

On the convergence and superconvergence for a class of two-dimensional time fractional reaction-subdiffusion equations

This paper analyzes a class of two-dimensional (2-D) time fractional reaction-subdiffusion equations with variable coefficients. The high-order L2-1_σ time-stepping scheme on graded meshes is presented to deal with the weak singularity at the initial time t = 0, and the bilinear finite element method (FEM) on anisotropic meshes is used for spatial discretization. Using the modified discrete fractional Grönwall inequality, and combining the interpolation operator and the projection operator, the L²-norm error estimation and H¹-norm superclose results are rigorously proved. The superconvergence result in the H¹-norm is derived by applying the interpolation postprocessing technique. Finally, numerical examples are presented to verify the validation of our theoretical analysis.