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The search for multivariate quadrature rules of minimal size with a specified polynomial accuracy has been the topic of many years of research. Finding such a rule allows accurate integration of moments, which play a central role in many aspects of scientific computing with complex models. The contribution of this paper is twofold. First, we provide novel mathematical analysis of the polynomial quadrature problem that provides a lower bound for the minimal possible number of nodes in a polynomial rule with specified accuracy. We give concrete but simplistic multivariate examples where a minimal quadrature rule can be designed that achieves this lower bound, along with situations that showcase when it is not possible to achieve this lower bound. Our second contribution is the formulation of an algorithm that is able to efficiently generate multivariate quadrature rules with positive weights on non-tensorial domains. Our tests show success of this procedure in up to 20 dimensions. We test our method on applications to dimension reduction and chemical kinetics problems, including comparisons against popular alternatives such as sparse grids, Monte Carlo and quasi Monte Carlo sequences, and Stroud rules. The quadrature rules computed in this paper outperform these alternatives in almost all scenarios. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the a priori error analysis for a convection-dominated diffusion 2D problem, when applying the HDG method on a family of anisotropic triangulations. It is known that in this case, boundary or interior layers may appear. Therefore, it is important to resolve these layers in order to recover, if possible, the expected order of approximation. In this work, we extend the use of HDG method on anisotropic meshes. In this context, when the discrete local spaces are polynomials of degree k ≥ 0 , this approach is able to recover an order of convergence k + 1 2 in L 2 for all the variables, under certain assumptions on the stabilization parameter and family of triangulations. Numerical examples confirm our theoretical results. Highlights • We develop an a priori error analysis for a HDG scheme defined on anisotropic meshes that are made of triangles. • We require that the family of triangulations satisfy the maximum angle condition, which is usual in this case. • We include numerical examples that validate our theoretical results. [ABSTRACT FROM AUTHOR]

The efficient solution of coupled PDEs/ODEs problems arising in cardiac electrophysiology is of key importance whenever interested to study the electrical behavior of the tissue for several instances of relevant physical and/or geometrical parameters. This poses significant challenges to reduced order modeling (ROM) techniques –such as the reduced basis method –traditionally employed when dealing with the repeated solution of parameter dependent differential equations. Indeed, the nonlinear nature of the problem, the presence of moving fronts in the solution, and the high sensitivity of this latter to parameter variations, make the application of standard ROM techniques very problematic. In this paper we propose a local ROM built through a k -means clustering in the state space of the snapshots for both the solution and the nonlinear term. Several comparisons among alternative local ROMs on a benchmark test case show the effectivity of the proposed approach. Finally, the application to a parametrized problem set on an idealized left-ventricle geometry shows the capability of the proposed ROM to face complex problems. [ABSTRACT FROM AUTHOR]

Current material point method (MPM) particle updates use a PIC approach, a FLIP approach, or a linear combination of PIC and FLIP. A PIC update filters velocity in each time step, which causes unwanted numerical diffusion, while FLIP eliminates that diffusion, but may retain too much noise. This paper develops a new particle update termed XPIC( m ) (for eXtended PIC of order m ) because it generalizes PIC updates. XPIC(1) is identical to current PIC methods, but higher orders of XPIC( m ) address the over filtering and numerical diffusion of PIC, while still filtering out noise caused by the nontrivial null space of the extrapolation matrix used in MPM. As m → ∞ , XPIC( m ) converges to a modified FLIP update with orthogonal removal of null space noise. The frequency response and filtering properties of XPIC( m ) are investigated and several numerical examples demonstrate its advantages over other update methods. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we are concerned with meshfree particle methods for the solutions of the classical plate model. The vertical displacement of a thin plate is governed by a fourth order elliptic equation and thus the approximation functions for numerical solutions are required to have continuous partial derivatives. Hence, the conventional finite element method has difficulties to solve the fourth order problems. Meshfree methods have the advantage of constructing smooth approximation functions, however, most of the earlier works on meshfree methods for plate problems used either moving least squares method with penalty method or coupling FEM with meshfree method to deal with essential boundary conditions. In this paper, by using generalized product partition of unity, introduced by Oh et al., we introduce meshfree particle methods in which approximation functions have high order polynomial reproducing property and the Kronecker delta property. We also prove error estimates for the proposed meshfree methods. Moreover, to demonstrate the effectiveness of our method, results of the proposed method are compared with existing results for various shapes of plates with variety of boundary conditions and loads. [Copyright &y& Elsevier]

Abstract: This paper discusses stabilization of an optimization procedure to determine an appropriate semivariogram parameter using the empirical semivariogram for Kriging-based approximation. From a viewpoint of a computational cost for constructing a surrogate model, a semivariogram fitting approach using the empirical semivariogram can be usable for the parameter estimation of a semivariogram function. However, instability of the optimization procedure for determination of the semivariogram parameter may be observed in some cases and it causes to generate an invalid surrogate model. For this problem, a simple technique for stabilization of the optimization for the parameter determination is proposed in this paper. The proposed approach employs a normalization technique of input data with respect to values of variables and an objective function. The proposed method is applied to some numerical examples, and the numerical results illustrate validity and effectiveness of the proposed method. [Copyright &y& Elsevier]

Abstract: We begin the mathematical study of the k-method utilizing the theory of Kolmogorov n-widths. The k-method is a finite element technique where spline basis functions of higher-order continuity are employed. It is a fundamental feature of the new field of isogeometric analysis. In previous works, it has been shown that using the k-method has many advantages over the classical finite element method in application areas such as structural dynamics, wave propagation, and turbulence. The Kolmogorov n-width and sup–inf were introduced as tools to assess the effectiveness of approximating functions. In this paper, we investigate the approximation properties of the k-method with these tools. Following a review of theoretical results, we conduct a numerical study in which we compute the n-width and sup–inf for a number of one-dimensional cases. This study sheds further light on the approximation properties of the k-method. We finish this paper with a comparison study of the k-method and the classical finite element method and an analysis of the robustness of polynomial approximation. [Copyright &y& Elsevier]

Abstract: In this paper, a finite-difference scheme for incompressible flow problems is treated. The scheme uses non-staggered grid for velocity approximation. A special stabilization is introduced to ensure the well-posedness and optimal approximation properties of the scheme. The stability estimate is proved in the form of a mesh-independent bound for the norm of discrete operator inverse. The finite-difference method is particularly suitable for problems in which velocity vector is involved in additional quantities that enter the system of flow equations as, for example, in the Bingham problem. We describe this application in the paper in some detail. Results of numerical experiments are included that confirm the applicability and optimality of the method. [Copyright &y& Elsevier]

Abstract: A spatial stabilization via bubble functions of linear finite element methods for nonlinear evolutionary convection–diffusion equations is discussed. The method of lines with SUPG discretization in space leads to numerical schemes that are not only difficult to implement, when considering nonlinear evolutionary equations, but also do not produce satisfactory results. The method we propose can be seen as an alternative to this kind of methods. Once the numerical approximation belonging to a linear finite element space enriched with bubble functions is obtained, the bubble part is discarded. The linear part is shown to give a stabilized approximation to the solution being approached. The bubble functions are deduced using a linear steady convection–diffusion model in such a way that the linear part of the approximation to the linear steady model (after static condensation of bubbles) reproduces the SUPG method in the convection-dominated regime. However, for the nonlinear evolutionary equations we consider in the paper the method we propose is not equivalent to the SUPG method. Some numerical experiments are provided in the paper to show the efficiency of the procedure. [Copyright &y& Elsevier]

Abstract: This paper considers the spectral element approximation of the Stokes problem and the conditioning of the resulting discrete problem. The well-posedness of the variational formulation of the Stokes problem and, in particular, the uniqueness of the pressure has been demonstrated when the subspace of square-integrable functions having vanishing mean is chosen for the pressure space. In the discrete setting a conforming subspace is chosen for the discrete pressure space and the analysis of the discrete problem is based on this choice. However, this space is not used in practical implementations of finite or spectral element methods for the Stokes problem. In this paper it is shown how the zero volume condition on the pressure may be accommodated within the trial space by modifying the continuity equation in a consistent manner. The dependence of the condition number of the Uzawa operator and the accuracy of the spectral element approximation on the weighting factor is investigated. [Copyright &y& Elsevier]

Non-conforming approximation methods are becoming increasingly popular because of the potential to apply to multi-material and multi-model analysis for both bounded and unbounded domains. In this paper, we present a least-square approximation based method to solve the one or two dimensional elliptic problems on an unbounded domain. The method gives exponential accuracy and shows superior performance when compared to other numerical methods. Differentiability estimates and the main stability estimate theorem, using a non-conforming spectral element method, are also discussed. The exponential convergence rate of the proposed method is also shown through rigorous error estimate and specific numerical examples. [ABSTRACT FROM AUTHOR]

It has been proven that the knowledge of an accurate approximation of the Dirichlet-to-Neumann (DtN) map is useful for a large range of applications in wave scattering problems. We are concerned in this paper with the construction of an approximate local DtN operator for time-harmonic elastic waves. The main contributions are the following. First, we derive exact operators using Fourier analysis in the case of an elastic half-space. These results are then extended to a general three-dimensional smooth closed surface by using a local tangent plane approximation. Next, a regularization step improves the accuracy of the approximate DtN operators and a localization process is proposed. Finally, a first application is presented in the context of the On-Surface Radiation Conditions method. The efficiency of the approach is investigated for various obstacle geometries at high frequencies. [ABSTRACT FROM AUTHOR]

In this paper a finite difference scheme is proposed for multidimensional convection-diffusion-reaction equations, particularly designed to treat the most interesting case of small diffusion. It is based closely on the work Şendur and Neslitürk (2011). Application of the method to multidimensional convection-diffusion-reaction equation is based on a simple splitting of the convection-diffusion-reaction equation and then joining their approximations obtained with Şendur and Neslitürk (2011). The method adapts very well to all regimes with continuous transitions from one regime to another. Numerical tests show good performance of the method and superiority with respect to well known stabilized finite element methods. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, some new generalized Newton’s methods for the resolution of elastostatic frictional contact problem approximated by finite elements are presented and compared to existing ones. A numerical experimentation is performed to compare the different methods, especially with respect to the sensitivity to the method parameter. Two different strategies to approximate the contact and friction condition are considered: a nodal and an integral one. Existence and uniqueness results of the solution to the discretized problem are also discussed. [Copyright &y& Elsevier]

Abstract: In this paper, we propose and analyze a new stabilized finite element method using continuous piecewise linear (or bilinear) elements for solving 2D reaction–convection–diffusion equations. The equation under consideration involves a small diffusivity and a large reaction coefficient , leading to high Péclet number and high Damköhler number. In addition to giving error estimates of the approximations in and norms, we explicitly establish the dependence of error bounds on the diffusivity, the norm of convection field, the reaction coefficient and the mesh size. Our analysis shows that the proposed method is particularly suitable for problems with a small diffusivity and a large reaction coefficient, or more precisely, with a large mesh Péclet number and a large mesh Damköhler number. Several numerical examples exhibiting boundary or interior layers are given to illustrate the high accuracy and stability of the proposed method. The results obtained are also compared with those of existing stabilization methods. [Copyright &y& Elsevier]

Abstract: This paper introduces a discontinuous method for the efficient determination of an approximate numerical solution of the two-dimensional advection–diffusion equation. Using the VTCR methodology (see ), this method involves free-space solutions of the governing partial differential equation. For the advection–diffusion equation with constant coefficients, the free-space solutions are exponential functions with a sharp gradient. The continuity of the solution across element boundaries is enforced weakly through a dedicated variational formulation. Preliminary results for a certain type of benchmark problem suggest that this approach is a promising numerical tool for handling such problems. [Copyright &y& Elsevier]

Abstract: Based on complex variable theory and moving least-squares (MLS) approximation, the improved complex variable moving least-squares (ICVMLS) approximation is discussed in this paper. Compared with complex variable moving least-squares (CVMLS) approximation, the function in the ICVMLS approximation has an explicit physics meaning. By using a new basis function, the ICVMLS approximation can obtain greater precision and computational efficiency. Based on the ICVMLS approximation, an improved complex variable element-free Galerkin (ICVEFG) method, which belongs to a novel element free Galerkin (EFG) method, is presented for two-dimensional large deformation problems. The Galerkin weak form is employed to obtain the equations, while the penalty method is used to apply the essential boundary conditions. Then the corresponding formulae of the ICVEFG method for two-dimensional large deformation problems are obtained. Compared with the EFG method, the ICVEFG method has greater precision and efficiency. [Copyright &y& Elsevier]

Abstract: In this paper we address the numerical approximation of the incompressible Navier–Stokes equations in a moving domain by the spectral element method and high order time integrators. We present the Arbitrary Lagrangian Eulerian (ALE) formulation of the incompressible Navier–Stokes equations and propose a numerical method based on the following kernels: a Lagrange basis associated with Fekete points in the spectral element method context, BDF time integrators, an ALE map of high degree, and an algebraic linear solver. In particular, the high degree ALE map is appropriate to deal with a computational domain whose boundary is described with curved elements. Finally, we apply the proposed strategy to a test case. [Copyright &y& Elsevier]

Abstract: In this paper we present new superconvergence results for the local discontinuous Galerkin (LDG) method applied to the second-order scalar wave equation in one space dimension. Numerical experiments show convergence rate for the LDG solution and O(h p+2) superconvergent solutions at Radau points. More precisely, a local error analysis reveals that, at a fixed time t, the leading terms of the discretization errors for the solution and its derivative using p-degree polynomial approximations are proportional to the (p +1)-degree right Radau and (p +1)-degree left Radau polynomials, respectively. Thus, the p-degree LDG solution is O(h p+2) superconvergent at the roots of the (p +1)-degree right Radau polynomial and the derivative of the LDG solution is O(h p+2) superconvergent at the roots of the (p +1)-degree left Radau polynomial. These results are used to construct simple, efficient, and asymptotically correct a posteriori error estimates in regions where solutions are smooth. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori errors estimates under mesh refinement. [Copyright &y& Elsevier]

Abstract: Generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries. The paper at hand introduces a B-spline finite element method, which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved. Its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement, which adds several levels of local basis functions in the close vicinity of interfaces, but unfitted to their exact location, so that a simple regular grid of knot span elements can be maintained. Numerical experiments show that an hp-refinement strategy, which simultaneously increases the polynomial degree of the B-spline basis and the levels of refinement around interfaces, achieves exponential rates of convergence despite the presence of discontinuities. It is also demonstrated that the hierarchical B-spline FEM can be used to transfer the recently introduced Finite Cell concept to geometrically nonlinear problems. Its computational performance, imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one, two and three dimensions, and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the phase separation on general surfaces by solving the nonlinear Cahn–Hilliard equation using a finite element method. A fully discrete approximation scheme is introduced, and we establish a priori estimates for the discrete solution that does not rely on any knowledge of the exact solution beyond the initial time. This in turn leads to convergence and optimal error estimates of the discretization scheme. Numerical examples are also provided to substantiate the theoretical results. [Copyright &y& Elsevier]

Abstract: We introduce in this paper a new direction splitting algorithm for solving the incompressible Navier–Stokes equations. The main originality of the method consists of using the operator (I − ∂ xx )(I − ∂ yy )(I − ∂ zz ) for approximating the pressure correction instead of the Poisson operator as done in all the contemporary projection methods. The complexity of the proposed algorithm is significantly lower than that of projection methods, and it is shown the have the same stability properties as the Poisson-based pressure-correction techniques, either in standard or rotational form. The first-order (in time) version of the method is proved to have the same convergence properties as the classical first-order projection techniques. Numerical tests reveal that the second-order version of the method has the same convergence rate as its second-order projection counterpart as well. The method is suitable for parallel implementation and preliminary tests show excellent parallel performance on a distributed memory cluster of up to 1024 processors. The method has been validated on the three-dimensional lid-driven cavity flow using grids composed of up to 2×109 points. [Copyright &y& Elsevier]

Abstract: In this paper, we study finite element approximations of a class of nonlinear eigenvalue problems arising from quantum physics. We derive both a priori and a posteriori finite element error estimates and obtain optimal convergence rates for both linear and quadratic finite element approximations. In particular, we analyze the convergence and complexity of an adaptive finite element method. In our analysis, we utilize certain relationship between the finite element eigenvalue problem and the associated finite element boundary value approximations. We also present several numerical examples in quantum physics that support our theory. [Copyright &y& Elsevier]

Abstract: This paper introduces an isogeometric framework for the numerical analysis of contact problems. In this approach, NURBS is utilized as basis for geometric representation and analysis. The contact surfaces and the underlying solids are parameterized in an identical paradigm. The use of NURBS geometry eliminates at the onset the geometric discontinuities induced by the traditional faceted approximation. Contact detection is carried out patch-wise, instead of element-wise, leading to an efficient and robust numerical algorithm. A strategy for smoothing sharp corners is also described. The algorithm does not rely on fillets but entails only a minor perturbation to the NURBS parameters. A variationally consistent surface-to-surface formulation is developed in the context of frictionless contact. Numerical examples are presented to demonstrate the performance and utility of the method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we present stabilized finite element methods to discretize in space the monochromatic radiation transport equation. These methods are based on the decomposition of the unknowns into resolvable and subgrid scales, with an approximation for the latter that yields a problem to be solved for the former. This approach allows us to design the algorithmic parameters on which the method depends, which we do here when the discrete ordinates method is used for the directional approximation. We concentrate on two stabilized methods, namely, the classical SUPG technique and the orthogonal subscale stabilization. A numerical analysis of the spatial approximation for both formulations is performed, which shows that they have a similar behavior: they are both stable and optimally convergent in the same mesh-dependent norm. A comparison with the behavior of the Galerkin method, for which a non-standard numerical analysis is done, is also presented. [ABSTRACT FROM AUTHOR]

Abstract: Accurate tangent stiffness matrices are essential for solving nonlinear structural problems using Newton-like methods. This paper presents a numerical procedure for generating accurate tangent stiffness matrices from the derivative approximation of internal force using the complex variable derivative method (CVDM). The present procedure is suitable for problems that require complex constitutive relations and/or kinematic nonlinearities whose models are difficult to linearize, as well as for the incorporation of Newton-like solution strategies with nonlinear structural analyses. [ABSTRACT FROM AUTHOR]

Abstract: We study subgrid artificial viscosity methods for approximating solutions to the Navier–Stokes equations. Two methods are introduced that add viscous stabilization via an artificial viscosity, then remove it only on a coarse mesh. These methods can be considered as conforming, mixed methods, the first for velocity and vorticity, and the second for velocity and its gradient, the former incorporating a naturally arising grad–div stabilization term. In this paper, we rigorously study the first scheme analytically, showing that it is unconditionally stable and optimally convergent, as well as both schemes computationally. Numerical experiments demonstrate the advantages of both of these methods. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we derive and analyze some a posteriori error estimators for the stabilized P1 nonconforming approximation of the Stokes problem involving the strain tensor. This will be done by decomposing the numerical error in a proper way into conforming and nonconforming contributions. The error estimator for the nonconforming error is obtained in the standard way, and the implicit error estimator for the conforming error is derived by applying the equilibrated residual method. A crucial part of this work is construction of approximate normal stresses on interelement boundaries which will serve as equilibrated Neumann data for local Stokes problems. It turns out that such normal stresses can be simply computed by local weak residuals of the discrete system plus jumps of the velocity solution and that a stronger equilibration condition is satisfied to ensure solvability of local Stokes problems. We also derive a simple explicit error estimator based on the nonsymmetric tensor recovery of the normal stress error. Numerical results are provided to illustrate the performance of our error estimators. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study from the numerical point of view a strain-adaptive bone remodelling that couples the displacements and the apparent density (the porosity) of the bone. The rate of this density at a particular location is described as an objective function, which depends on a particular stimulus at that location. The variational problem is written as a coupled system of a nonlinear variational equation for the displacement field and a nonlinear parabolic variational inequality for the apparent density. Then, fully discrete approximations are provided by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are proved, from which, under adequate regularity conditions, the linear convergence of the algorithm is deduced. Finally, some numerical simulations involving one- and two-dimensional test examples are presented to demonstrate the accuracy of the approximation and the behaviour of the solution. [Copyright &y& Elsevier]

Abstract: In this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by semilinear parabolic equations. The space discretization of the state variable is done using usual mixed finite elements, whereas the time discretization is based on difference methods. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. By applying some error estimates techniques of mixed finite element methods, we derive a priori error estimates both for the coupled state and the control approximation. Finally, we present a numerical example which confirms our theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, a new numerical approach, based on mortar discretization of interface fields, is proposed to solve lubricated contact problems between deformable solid bodies. The Reynolds approximation to the thin film fluid equations is used to describe the fluid behavior on the interface, while the equations of small strain elasticity (to which finite elements are applied) are assumed to govern the solid body behavior. The result of this approach is a monolithic scheme in which the fluid and solid mechanics are solved simultaneously, without the need for iterative or staggered approaches between the fluid and solid phases. Key features of the approach are that the fluid film thickness, directly related to the deformation of the solid phase, is computed from a least squares projection based on dual basis functions (as pioneered by Wohlmuth [B.I. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition, Springer-Verlag, Heidelberg, 2001]). The free boundary problem for the fluid phase, corresponding to the boundary between lubricated and cavitated regions, is regularized with a penalty method and thus resolved as part of the solution. Several numerical examples are given to demonstrate the approach. [Copyright &y& Elsevier]

Abstract: In this paper, we study the numerical approximation of bedload sediment transport due to shallow layer flows. The hydrodynamical component is modeled by a 2D shallow water system and the morphodynamical component by a solid transport discharge formula that depends on the hydrodynamical variables. The coupled system can be written as a nonconservative hyperbolic system. To discretize it, first we consider a Roe-type first order scheme as well as a variant based on the use of flux limiters. These first order schemes are then extended to second order accuracy by means of a new MUSCL-type reconstruction operator on unstructured meshes. Finally, some numerical tests are presented. [Copyright &y& Elsevier]

Abstract: This paper describes new strict error bounds of calculated outputs of interest in linear dynamics. All sources of errors, including modeling errors, are taken into account, which makes such error bounds suitable tools for analyzing various approximations in dynamics. Small-displacement problems are included through the classical thermodynamics framework involving internal state variables. Preliminary numerical illustrations are given. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the hp-adaptive multilevel solution of second order elliptic partial differential equations using the meshfree particle–partition of unity method. The proposed refinement scheme automatically constructs new discretization points (or particles), the meshfree analogue of an adaptive h-refinement, and local approximation spaces with better local resolution, a p-refinement. The refinement process is steered with the help of an a-posteriori subdomain-type error indicator. We present results of numerical experiments in two and three space dimensions which demonstrated the overall efficiency of the proposed scheme. [Copyright &y& Elsevier]

This paper presents an adaptive Discontinuous-Galerkin finite element method to solve linear advection–reaction equations. The method was developed for the prediction of hemolysis in prosthetic devices, where the flow may include recirculation zones. To ensure the well-posedness of the problem, we solve the transient form of the PDE. Time discretization is achieved with an implicit Euler scheme in order to obtain the steady-state solution rapidly. Linear interpolation functions are used to approximate the hemolysis field. The a posteriori estimation of the error is obtained by solving an advection–reaction equation for the error, which is approximated with quadratic interpolation functions. Then, elementary estimations of the error are computed using a semi-norm developed to achieve the best possible results, even in the presence of recirculation. Code verification – to assess the convergence and conservation properties of the method – is performed with the method of manufactured solutions and with a conservation test using a sharp gaussian function as a reaction term. Excellent results are obtained in both cases. Predictions of hemolysis in a dialysis cannula are presented as an engineering application of our method. [Copyright &y& Elsevier]

Abstract: In this paper, we develop a nonconforming finite element methodology using a three-field formulation to analyze a fluid–structure interaction problem. The methodology is used to couple a Lagrangian model describing the structure with the arbitrary Lagrangian–Eulerian strategy used to describe the fluid in order to simulate a full unsteady physical phenomenon. Consistency error estimates are obtained which show that the numerical scheme employed yields a first order approximation for the solution to the fluid–structure interaction problem. Finally, we present a discrete energy estimate to demonstrate the stablity of the proposed method. [Copyright &y& Elsevier]

Abstract: This paper develops an approximate theory, which combines the volume incompressible condition and Levy–Mises equation, to calculate preform shape, nosing ratio, and loading rate in the tube nosing process by spherical die. An elasto-plastic finite element method based on the updated Lagrangian formulation is used to verify the nosing ratio, tube end strain rate and thickness distribution at the preform site after nosing. From comparison of the results obtained in the theory and FEM, the relations of nosing ratio and tube end strain rate to punch stroke ratio, shows good agreement between the approximate theory and finite element prediction. A uniform thickness distribution, with small deviation form the expected thickness, was obtained for the preform site after nosing process. It proves that the analytical expressions will be able to provide a good reference guide for the tube preform design in the nosing forming. [Copyright &y& Elsevier]

In this paper and its sequels, we introduce and analyze a new class of methods, collectively called the reproducing kernel element method (RKEM). The central idea in the development of the new method is to combine the strengths of both finite element methods (FEM) and meshfree methods. Two distinguished features of RKEM are: the arbitrarily high order smoothness and the interpolation property of the shape functions. These properties are desirable especially in solving Galerkin weak forms of higher order partial differential equations and in treating Dirichlet boundary conditions. So unlike the FEM, there is no need for special treatment with the RKEM in solving high order equations. Compared to meshfree methods, Dirichlet boundary conditions do not present any difficulty in using the RKEM. A rigorous error analysis and convergence study of the method are presented. The performance of the method is illustrated and assessed through some numerical examples. [Copyright &y& Elsevier]