Multiphysics mixed finite element method with Nitsche's technique for Stokes-poroelasticity problem

In this article, we propose a multiphysics mixed finite element method with Nitsche's technique for Stokes-poroelasticity problem. Firstly, we reformulate the poroelasticity part of the original problem by introducing two pseudo-pressures to into a “fluid–fluid” coupled problem so that we can use the classical stable finite element pairs to deal with this problem conveniently. Then, we prove the existence and uniqueness of weak solution of the reformulated problem. And we use Nitsche's technique to approximate the coupling condition at the interface to propose a loosely-coupled time-stepping method to solve three subproblems at each time step–a Stokes problem, a generalized Stokes problem and a mixed diffusion problem. And the proposed method does not require any restriction on the choice of the discrete approximation spaces on each side of the interface provided that appropriate quadrature methods are adopted. Also, we give the stability analysis and error estimates of the loosely-coupled time-stepping method. Finally, we give the numerical tests to show that the proposed numerical method has a good stability and no “locking” phenomenon.     

Numerical integration in Galerkin meshless methods, applied to elliptic Neumann problem with non-constant coefficients

In this paper, we explore the effect of numerical integration on the Galerkin meshless method used to approximate the solution of an elliptic partial differential equation with non-constant coefficients with Neumann boundary conditions. We considered Galerkin meshless methods with shape functions that reproduce polynomials of degree k?≥?1. We have obtained an estimate for the energy norm of the error in the approximate solution under the presence of numerical integration. This result has been established under the assumption that the numerical integration rule satisfies a certain discrete Green’s formula, which is not problem dependent, i.e., does not depend on the non-constant coefficients of the problem. We have also derived numerical integration rules satisfying the discrete Green’s formula.     

A Well-Conditioned,Nonconforming Nitsche's Extended Finite Element Method for Elliptic Interface Problems

In this paper, we introduce a nonconforming Nitsche's extended finite element method (NXFEM) for elliptic interface problems on unfitted triangulation elements. The solution on each side of the interface is separately expanded in the standard nonconforming piecewise linear polynomials with the edge averages as degrees of freedom. The jump conditions on the interface and the discontinuities on the cut edges (the segment of edges cut by the interface) are weakly enforced by the Nitsche's approach. In the method, the harmonic weighted fluxes are used and the extra stabilization terms on the interface edges and cut edges are added to guarantee the stability and the well conditioning. We prove that the convergence order of the errors in energy and $L^2$ norms are optimal. Moreover, the errors are independent of the position of the interface relative to the mesh and the ratio of the discontinuous coefficients. Furthermore, we prove that the condition number of the system matrix is independent of the interface position. Numerical examples are given to confirm the theoretical results.     

Local radial basis function interpolation method to simulate 2D fractional‐time convection‐diffusion‐reaction equations with error analysis

In this article, a kind of meshless local radial point interpolation (MLRPI) method is proposed to two‐dimensional fractional‐time convection‐diffusion‐reaction equations and satisfactory agreements are archived. This method is based on meshless methods and benefits from collocation ideas but it does not belong to the traditional global meshless collocation methods. In MLRPI method, it does not need any kind of integration locally or globally over small quadrature domains which is essential in the finite element method and those meshless methods based on Galerkin weak form. Also, it is not needed to determine shape parameter which plays important role in collocation method based on the radial basis functions (Kansa's method). Therefore, computational costs of this kind of MLRPI method is less expensive. The stability and convergence of this meshless approach are discussed and theoretically proven. It is proved that the present meshless formulation is very effective for modeling and simulation of fractional differential equations. Furthermore, the numerical studies on sensitivity analysis and convergence analysis show the stability and reliable rates of convergence. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 974–994, 2017     

Análise de vigas laminadas utilizando o natural neighbour radial point interpolation method

In this work, a meshless method, “natural neighbour radial point interpolation method” (NNRPIM), is applied to the one‐dimensional analysis of laminated beams, considering the theory of Timoshenko.The NNRPIM combines the mathematical concept of natural neighbours with the radial point interpolation. Voronoï diagrams allows to impose the nodal connectivity and the construction of a background mesh for integration purposes, via influence cells. The construction of the NNRPIM interpolation functions is shown, and, for this, it is used the multiquadratic radial basis function. The generated interpolation functions possess infinite continuity and the delta Kronecker property, which facilitates the enforcement of boundary conditions, since these can be directly imposed, as in the finite element method (FEM).In order to obtain the displacements and the deformation fields, it is considered the Timoshenko theory for beams under transverse efforts. Several numerical examples of isotropic beams and laminated beams are presented in order to demonstrate the convergence and accuracy of the proposed application. The results obtained are compared with analytical solutions available in the literature.     

Theoretical analysis of numerical integration in Galerkin meshless methods

In this paper, we study effects of numerical integration on Galerkin meshless methods for solving elliptic partial differential equations with Neumann boundary conditions. The shape functions used in the meshless methods reproduce linear polynomials. The numerical integration rules are required to satisfy the so-called zero row sum condition of stiffness matrix, which is also used by Babuška et al. (Int. J. Numer. Methods Eng. 76:1434–1470, 2008). But the analysis presented there relies on a certain property of the approximation space, which is difficult to verify. The analysis in this paper does not require this property. Moreover, the Lagrange multiplier technique was used to handle the pure Neumann condition. We have also identified specific numerical schemes, diagonal elements correction and background mesh integration, that satisfy the zero row sum condition. The numerical experiments are carried out to verify the theoretical results and test the accuracy of the algorithms.     

Spectral meshless radial point interpolation (SMRPI) method to two‐dimensional fractional telegraph equation

H. Ammari In this article, an innovative technique so‐called spectral meshless radial point interpolation (SMRPI) method is proposed and, as a test problem, is applied to a classical type of two‐dimensional time‐fractional telegraph equation defined by Caputo sense for (1 < α≤2). This new methods is based on meshless methods and benefits from spectral collocation ideas, but it does not belong to traditional meshless collocation methods. The point interpolation method with the help of radial basis functions is used to construct shape functions, which play as basis functions in the frame of SMRPI method. These basis functions have Kronecker delta function property. Evaluation of high‐order derivatives is not difficult by constructing operational matrices. In SMRPI method, it does not require any kind of integration locally or globally over small quadrature domains, which is essential of the finite element method (FEM) and those meshless methods based on Galerkin weak form. Also, it is not needed to determine strict value for the shape parameter, which plays an important role in collocation method based on the radial basis functions (Kansa's method). Therefore, computational costs of SMRPI method are less expensive. Two numerical examples are presented to show that SMRPI method has reliable rates of convergence. Copyright © 2015 John Wiley & Sons, Ltd.     

Subregion-Adaptive Integration of Functions Having a Dominant Peak

Abstract

Many statistical multiple integration problems involve integrands that have a dominant peak. In applying numerical methods to solve these problems, statisticians have paid relatively little attention to existing quadrature methods and available software developed in the numerical analysis literature. One reason these methods have been largely overlooked, even though they are known to be more efficient than Monte Carlo for well-behaved problems of low dimensionality, may be that when applied naively they are poorly suited for peaked-integrand problems. In this article we use transformations based on “split t” distributions to allow the integrals to be efficiently computed using a subregion-adaptive numerical integration algorithm. Our split t distributions are modifications of those suggested by Geweke and may also be used to define Monte Carlo importance functions. We then compare our approach to Monte Carlo. In the several examples we examine here, we find subregion-adaptive integration to be substantially more efficient than importance sampling.     

The propensity to disrupt and the disruption nucleolus of a characteristic function game

Gately [1974] recently introduced the concept of an individual player's “propensity to disrupt” a payoff vector in a three-person characteristic function game. As a generalisation of this concept we propose the “disruption nucleolus” of ann-person game. The properties and computational possibilities of this concept are analogous to those of the nucleolus itself. Two numerical examples are given.     

A meshless based numerical technique for traveling solitary wave solution of Boussinesq equation

In this paper, we employ the boundary-only meshfree method to find out numerical solution of the classical Boussinesq equation in one dimension. The proposed method in the current paper is a combination of boundary knot method and meshless analog equation method. The boundary knot technique is an integration free, boundary-only, meshless method which is used to avoid the known disadvantages of the method of fundamental solution. Also, we use the meshless analog equation method to replace the nonlinear governing equation with an equivalent nonhomogeneous linear equation. A predictor-corrector scheme is proposed to solve the resulted differential equation of the collocation. The numerical results and conclusions are obtained for both the ‘good’ and the ‘bad’ Boussinesq equations.     

The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations

In this paper, we address some fundamental issues concerning “time marching” numerical schemes for computing steady state solutions of boundary value problems for nonlinear partial differential equations. Simple examples are used to illustrate that even theoretically convergent schemes can produce numerical steady state solutions that do not correspond to steady state solutions of the boundary value problem. This phenomenon must be considered in any computational study of nonunique solutions to partial differential equations that govern physical systems such as fluid flows. In particular, numerical calculations have been used to “suggest” that certain Euler equations do not have a unique solution. For Burgers' equation on a finite spatial interval with Neumann boundary conditions the only steady state solutions are constant (in space) functions. Moreover, according to recent theoretical results, for any initial condition the corresponding solution to Burgers' equation must converge to a constant as t → ∞. However, we present a convergent finite difference scheme that produces false nonconstant numerical steady state “solutions.” These erroneous solutions arise out of the necessary finite floating point arithmetic inherent in every digital computer. We suggest the resulting numerical steady state solution may be viewed as a solution to a “nearby” boundary value problem with high sensitivity to changes in the boundary conditions. Finally, we close with some comments on the relevance of this paper to some recent “numerical based proofs” of the existence of nonunique solutions to Euler equations and to aerodynamic design.     

Dynamic analysis of porous materials: Numerical simulation with an adaptive space-time FEM

We investigated a space-time Galerkin method applied to the dynamic analysis of fully-saturated porous material. The governing set of equations are derived from the well-studied Theory of Porous Media (TPM). The numerical scheme consists of a coupled finite element discretization in the space-time domain. Discontinuous approximations of the primary variables in time are employed. A natural flux treatment is applied in time to impose the consistency between the adjacent time intervals. A simple space-time decoupled adaptive strategy based on the “jumps” in time and the ZZ error indicator in space is investigated. Numerical experiments demonstrate the efficiency and reliability of the proposed approach. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis,modeling, emergence & integration in complex systems: A modeling and integration framework & system biology

Analysis and Modeling is the first “phase” of understanding or developing a system. It is also, maybe more importantly, the foundation of understanding a natural science or system. It's abstract and conceptually difficult but, being foundational, contributes the most to the quality of understanding of (designed or natural) systems. Complex Systems have a natural hierarchy of levels and multiple subsystems. The character and functionality of each level or subsystem “emerges” across its boundaries. Both sides of these boundaries must be understood within that side's natural thought patterns. Integrated interdisciplinary collaboration is essential for making sense of complex systems; but collaboration among disciplines is difficult, because of their different ways of thinking. This creates a dilemma, “understanding complex systems” is one horn; “integrated interdisciplinary collaboration” is the other. This dilemma in complex system analysis/modeling and interdiscipline collaboration, is currently addressed by “grabbing the bull by the horns.” This takes on this doubly complex problem, by painstakingly building up abstract “bull wrestling” skills in and across domains and disciplines. There's another wrinkle; complexity requires interdisciplinary collaboration at deeper, more dissimilar, levels. The usual approach, finding a way to “pass between the horns of the dilemma” will not work here, due to this cross coupling. Rather than trying to pass between the horns, by abstracting away the coupling, we overtly organizing this coupling. We weave a semantic unification space of conceptual connections linking each side of a boundary to its appropriate way of thinking. This allows us to abstracting away the dilemma and iron out the wrinkle. The threads of common image schemas, cognitive metaphors and conceptual interfaces, weave a bridge between the semantics foundations and organizations of each problem. These allow addressing the problems synergistically. This paper presents and explores a naturally valid way for discipline specific and discipline integrating addressing complex systems. We start with the methodological insights from analysis and modeling from the perspective of object orientation, with its ontologies, organizing lexical semantics. We advance from there by integrating in imagistic, imaginative semantics and affordance based interaction methodology, as the keys to addressing complex systems analysis, modeling and integration. © 2007 Wiley Periodicals, Inc. Complexity, 2007     

Cultural Influences on Children's Probabilistic Thinking

This study investigated selected cultural influences on probabilistic thinking of 11–12-year-old children in England. Language, beliefs and experience are shown to influence the children's “informal knowledge” of probability, i.e., the intuitive knowledge they bring to school and use in thinking about probabilistic situations presented in school. Some of the pupils' responses in interviews and in a questionnaire were consistent with the “outcome approach” and with the use of certain heuristics: “representativeness”; “availability”; “equiprobability.” A significant proportion of pupils revealed superstitions. Standard “random devices” such as dice were regarded by some children as subject to personal, religious or causative influences. In a comparison between two culturally-contrasted subgroups in the same school, we found significant differences in scores on probability tests, even when taking account of numerical and non-verbal ability. But the differences between the two groups could almost entirely be explained by differences in language ability.     

Theoretical Analysis of the Reproducing Kernel Gradient Smoothing Integration Technique in Galerkin Meshless Methods

Numerical integration poses greater challenges in Galerkin meshless methods than finite element methods owing to the non-polynomial feature of meshless shape functions. The reproducing kernel gradient smoothing integration (RKGSI) is one of the optimal numerical integration techniques in Galerkin meshless methods with minimum integration points. In this paper, properties, quadrature rules and the effect of the RKGSI on meshless methods are analyzed. The existence, uniqueness and error estimates of the solution of Galerkin meshless methods under numerical integration with the RKGSI are established. A procedure on how to choose quadrature rules to recover the optimal convergence rate is presented.     

A meshless local moving Kriging method for two-dimensional solids

An improved meshless local Petrov-Galerkin method (MLPG) for stress analysis of two-dimensional solids is presented in this paper. The MLPG method based on the moving least-squares approximation is one of the recent meshless approaches. However, accurate imposition of essential boundary conditions in the MLPG method often presents difficulties because the MLPG shape functions does not possess the Kronecker delta property. In order to eliminate this shortcoming, this approach uses the moving Kriging interpolation instead of the traditional moving least-square approximation to construct the MLPG shape functions, and then, the Heaviside step function is used as the test function over a local sub-domain. In this method, the essential boundary conditions can be enforced as the FEM, no domain integration is needed and only regular boundary integration is involved. In addition, the sensitivity of several important parameters of the present method is mainly studied and discussed. Comparing with the original meshless local Petrov-Galerkin method, the present method has simpler numerical procedures and lower computation cost. The effectiveness of the present method for two-dimensional solids problem is investigated by numerical examples in this paper.     

A Laplace transform certified reduced basis method; application to the heat equation and wave equation

We present a certified reduced basis (RB) method for the heat equation and wave equation. The critical ingredients are certified RB approximation of the Laplace transform; the inverse Laplace transform to develop the time-domain RB output approximation and rigorous error bound; a (Butterworth) filter in time to effect the necessary “modal” truncation; RB eigenfunction decomposition and contour integration for Offline–Online decomposition. We present numerical results to demonstrate the accuracy and efficiency of the approach.     

Characteristic-based schemes for dispersive waves I. The method of characteristics for smooth solutions

In order to embark on the development of numerical schemes for stiff problems, we have studied a model of relaxing heat flow. To isolate those errors unavoidably associated with discretization, a method of characteristics is developed, containing three free parameters depending on the stiffness ratio. It is shown that such “decoupled” schemes do not take into account the interaction between the wave families and hence result in incorrect wave speeds. We also demonstrate that schemes can differ by up to two orders of magnitude in their rms errors even while maintaining second-order accuracy. We show that no method of characteristics solution can be better than second-order accurate. Next, we develop “coupled” schemes which account for the interactions, and here we obtain two additional free parameters. We demonstrate how coupling of the two wave families can be introduced in simple ways and how the results are greatly enhanced by this coupling. Finally, numerical results for several decoupled and coupled schemes are presented, and we observe that dispersion relationships can be a very useful qualitative tool for analysis of numerical algorithms for dispersive waves. © 1993 John Wiley & Sons, Inc.     

A methodology for solving chemical equilibrium systems

This paper describes a methodology for solving chemical equilibrium systems. The methodology is especially appropriate when many systems with the same structure must be solved quickly, as in finite-difference models of fluid flow and combustion. The approach features identifying a “canonical form” for such systems and exploiting this canonical form to effect a preliminary algebraic reduction. Then numerical scaling and iterative solution techniques are evoked. Both homotopy continuation and a variant of Newton's method are effective solution methods. This methodology is a significant advance over previous approaches to solving chemical equilibrium systems. The problem of solving such systems has heretofore been regarded as challenging; solution techniques have tended to emphasize case-by-case analyses and hit-or-miss numerical methods. We show that the problem can be solved by a unified approach, quickly and reliably.