A Fully Implicit Computational Strategy for Strongly Coupled Fluid–Solid Interaction

This article summarises the authors’ research work in the area of computational modelling of interaction of fluid flow with solid structures. Our approach relies on a fully implicit iterative solution strategy which resolves the strong coupling and allows for optimal rate of convergence of the residuals. Therefore, the methodology is a viable competitor for the solution of the highly nonlinear interaction of fluid flow with solid structures that experience large displacements and deformations. The key ingredients of our strategy include the following: Stabilised low order velocity–pressure finite elements are used for the modelling of the fluid flow combined with an arbitrary Lagrangian–Eulerian (ALE) strategy. For the temporal discretisation of both fluid and solid bodies, the discrete implicit generalised-α method is employed. An important aspect of the present work is the introduction of the independent interface discretisation, which allows an efficient, modular and expandable implementation of the solution strategy. A simple data transfer strategy based on a finite element type interpolation of the interface degrees of freedom guarantees kinematic consistency and equilibrium of the stresses along the interface. The resulting strongly coupled set of nonlinear equations is solved by means of a partitioned solution procedure, which is based on the Newton–Raphson methodology and incorporates the full linearisation of the overall incremental problem. Thus, asymptotically quadratic convergence of the residuals is achieved. Numerical examples are presented to demonstrate the robustness and efficiency of the methodology. Finally, we present the results obtained by combining the presented methodology with a remeshing procedure.     

Substructuring of a Signorini-type problem and Robin’s method for the Richards equation in heterogeneous soil

We prove a substructuring result for variational inequalities. It concerns but is not restricted to the Richards equation in heterogeneous soil, and it includes boundary conditions of Signorini’s type. This generalizes existing results for the linear case and leads to interface conditions known from linear variational equalities: continuity of Dirichlet and flux values in a weak sense. In case of the Richards equation, these are the continuity of the physical pressure and of the water flux, which is hydrologically reasonable. We use these interface conditions in a heterogeneous problem with piecewise constant soil parameters, which we address by the Robin method. We prove that, for a certain time discretization, the homogeneous problems in the subdomains including Robin and Signorini-type boundary conditions can be solved by convex minimization. As a consequence, we are able to apply monotone multigrid in the discrete setting as an efficient and robust solver for the local problems. Numerical results demonstrate the applicability of our approach.     

A numerical solution of the stochastic discrete algebraic Riccati equation

This article proposes two algorithms for solving a stochastic discrete algebraic Riccati equation which arises in a stochastic optimal control problem for a discrete-time system. Our algorithms are generalized versions of Hewer’s algorithm. Algorithm I has quadratic convergence, but needs to solve a sequence of extended Lyapunov equations. On the other hand, Algorithm II only needs solutions of standard Lyapunov equations which can be solved easily, but it has a linear convergence. By a numerical example, we shall show that Algorithm I is superior to Algorithm II in cases of large dimensions. This work was presented in part at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008     

A nonconforming finite element method¶with anisotropic mesh grading¶for the incompressible Navier–Stokes equations¶in domains with edges

Any solution of the incompressible Navier–Stokes equations in three-dimensional domains with edges has anisotropic singular behaviour which is treated numerically by using anisotropic finite element meshes. The velocity is approximated by Crouzeix–Raviart (nonconforming 𝒫₁) elements and the pressure by piecewise constants. This method is stable for general meshes since the inf-sup condition is satisfied without minimal or maximal angle condition. The existence of solutions to the discrete problems follows. Consistency error estimates for the divergence equation are obtained for anisotropic tensor product meshes. As applications, the consistency error estimate for the Navier–Stokes solution and some discrete Sobolev inequalities are derived on such meshes. These last results provide optimal error estimates in the uniqueness case by the use of appropriately refined anisotropic tensor product meshes, namely, if N _e is the number of elements of the mesh, we prove that the optimal order of convergence h∼ N _e ^{− 1/3}. Received:July 2001 / Accepted: July 2002     

Convergence of an upwind control-volume mixed finite element method for convection–diffusion problems

Summary We consider a upwind control volume mixed finite element method for convection–diffusion problem on rectangular grids. These methods use the lowest order Raviart–Thomas mixed finite element space as the trial functional space and associate control-volumes, or covolumes, with the vector variable as well as the scalar variable. Chou et al. [6] established a one-half order convergence in discrete L ²-norms. In this paper, we establish a first order convergence for both the vector variable as well as the scalar variable in discrete L ²-norms.      

A conservative and monotone mixed-hybridized finite element approximation of transport problems in heterogeneous domains

In this article, we discuss the numerical approximation of transport phenomena occurring at material interfaces between physical subdomains with heterogenous properties. The model in each subdomain consists of a partial differential equation with diffusive, convective and reactive terms, the coupling between each subdomain being realized through an interface transmission condition of Robin type. The numerical approximation of the problem in the two-dimensional case is carried out through a dual mixed-hybridized finite element method with numerical quadrature of the mass flux matrix. The resulting method is a conservative finite volume scheme over triangular grids, for which a discrete maximum principle is proved under the assumption that the mesh is of Delaunay type in the interior of the domain and of weakly acute type along the domain external boundary and internal interface. The stability, accuracy and robustness of the proposed method are validated on several numerical examples motivated by applications in biology, electrophysiology and neuroelectronics.     

Accelerated domain decomposition iterative procedures for mixed methods based on Robin transmission conditions

A nonoverlapping domain decomposition iterative procedure based on Robin transmission conditions applicable to elliptic boundary problems was first introduced by P.~L.~Lions and later discussed by a number of authors under the assumption that the weighting of the flux and the trace of the solution in the Robin interface condition be independent of the iterative step number. Recently, the authors [6] studied a finite difference method for a Dirichlet problem and introduced a cycle of weights for the flux in this interface condition and proved that an acceleration in the convergence rate similar to that occurring for alternating-direction iteration using a cycle of pseudo-time steps results. The objects of this paper are to describe an analogous procedure for a mixed finite element approximation for a model Neumann problem and to consider an overlapping subdomain of the iteration, while retaining the variable parameter cycle. It will be shown that a greater acceleration of the iteration can be obtained by combining overlap and the parameter cycle than by the separate use of either. Received: November 1997 / Revised version: December 1997     

A finite element based level set method for two-phase incompressible flows

We present a method that has been developed for the efficient numerical simulation of two-phase incompressible flows. For capturing the interface between the phases the level set technique is applied. The continuous model consists of the incompressible Navier–Stokes equations coupled with an advection equation for the level set function. The effect of surface tension is modeled by a localized force term at the interface (so-called continuum surface force approach). For spatial discretization of velocity, pressure and the level set function conforming finite elements on a hierarchy of nested tetrahedral grids are used. In the finite element setting we can apply a special technique to the localized force term, which is based on a partial integration rule for the Laplace–Beltrami operator. Due to this approach the second order derivatives coming from the curvature can be eliminated. For the time discretization we apply a variant of the fractional step θ-scheme. The discrete saddle point problems that occur in each time step are solved using an inexact Uzawa method combined with multigrid techniques. For reparametrization of the level set function a new variant of the fast marching method is introduced. A special feature of the solver is that it combines the level set method with finite element discretization, Laplace–Beltrami partial integration, multilevel local refinement and multigrid solution techniques. All these components of the solver are described. Results of numerical experiments are presented.     

Multichannel image segmentation using adaptive finite elements

We present an adaptive finite element algorithm for segmentation with denoising of multichannel images in two dimensions, of which an extension to three dimensional images is straight forward. It is based on a level set formulation of the Mumford–Shah approach proposed by Chan and Vese in (JVCIR 11:130–141,(2000); IEEE Trans Image Proces 10(2):266–277, (2001); Int J Comp Vis 50(3):271–293, (2002)) In case of a minimal partition problem an exact solution is given and convergence of the discrete solution towards this solution is numerically verified.     

The method of successive approximation applied to find the probability for an insurance company with random premiums

A risk process that describes the evolution of the capital of an insurance company is analyzed, random premiums and claims being available. Integral equations of nonbankruptcy probability as a function of the initial capital are derived. Necessary and sufficient conditions for the existence and uniqueness of solutions of these integral equations, and convergence conditions for the method of successive approximation for finding their solutions are established __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 112–127, January–February 2006.     

A Fast Solver for Boundary Integral Equations of the Modified Helmholtz Equation

The main purpose of this paper is to develop a fast fully discrete Fourier–Galerkin method for solving the boundary integral equations reformulated from the modified Helmholtz equation with boundary conditions. We consider both the nonlinear and the Robin boundary conditions. To tackle the difficulties caused by the two types of boundary conditions, we provide an improved version of the Galerkin method based on the Fourier basis. By employing a matrix compression strategy and efficient numerical quadrature schemes for oscillatory integrals, we obtain fully discrete nonlinear or linear system. Finally, we use the multilevel augmentation method to solve the resulting systems. We point out that the proposed method enjoys an optimal convergence order and a nearly linear computational complexity. The theoretical estimates are confirmed by the performance of this method on several numerical examples.     

A method for numerical solution of a multidimensional convection-diffusion problem

We propose a modification of the additive splitting algorithm to solve the convection-diffusion problem using an efficient finite-difference scheme. The modification decreases the number of data exchanges and their amount during the numerical solution of a system of multidimensional equations. Approximation, stability, and convergence are considered. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 100–107, January–February 2009.     

A Stable and Convergent Hodge Decomposition Method for Fluid–Solid Interaction

Fluid–solid interaction has been a challenging subject due to their strong nonlinearity and multidisciplinary nature. Many of the numerical methods for solving FSI problems have struggled with non-convergence and numerical instability. In spite of comprehensive studies, it has still been a challenge to develop a method that guarantees both convergence and stability. Our discussion in this work is restricted to the interaction of viscous incompressible fluid flow and a rigid body. We take the monolithic approach by Gibou and Min (J Comput Phys 231:3245–3263, 2012) that results in an augmented Hodge projection. The projection updates not only the fluid vector field but also the solid velocities. We derive the equivalence between the augmented Hodge projection and the Poisson equation with non-local Robin boundary condition. We prove the existence, uniqueness, and regularity for the weak solution of the Poisson equation, through which the Hodge projection is shown to be unique and orthogonal. We also show the stability of the projection in the sense that the projection does not increase the total kinetic energy of the fluid or the solid. Finally, we discuss a numerical method as a discrete analogue to the Hodge projection, then we show that the unique decomposition and orthogonality also hold in the discrete setting. As one of our main results, we prove that the numerical solution is convergent with at least first-order accuracy. We carry out numerical experiments in two and three dimensions, which validate our analysis and arguments.     

Domain Decomposition Algorithms for Two Dimensional Linear Schrödinger Equation

This paper deals with two domain decomposition methods for two dimensional linear Schrödinger equation, the Schwarz waveform relaxation method and the domain decomposition in space method. After presenting the classical algorithms, we propose a new algorithm for the Schrödinger equation with constant potential and a preconditioned algorithm for the general Schrödinger equation. These algorithms are then studied numerically. The numerical experiments show that the new algorithms can improve the convergence rate and reduce the computation time. Besides of the traditional Robin transmission condition, we also propose to use a newly constructed absorbing condition as the transmission condition.     

On the numerical solution of nonlinear systems of Volterra integro-differential equations with delay arguments

Particular cases of nonlinear systems of delay Volterra integro-differential equations (denoted by DVIDEs) with constant delay τ > 0, arise in mathematical modelling of ‘predator–prey’ dynamics in Ecology. In this paper, we give an analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for systems of this type. Then, from the perspective of applied mathematics, we consider the Volterra’s integro-differential system of ‘predator–prey’ dynamics arising in Ecology. We analyze the numerical issues of the introduced collocation method applied to the ‘predator–prey’ system and confirm that we can achieve the expected theoretical orders of convergence.      

Flux-free Finite Element Method with Lagrange Multipliers for Two-fluid Flows

In this paper we consider the flux-free finite element method based on the Eulerian framework for immiscible incompressible two-fluid flows, which is defined so as to preserve the mass of each fluid. This method is derived from the variational formulation including the flux-free constraint for the Navier–Stokes equations by the Lagrange multiplier technique. Focusing on the stationary problem, we prove the well-posedness of the finite element solution by a discrete inf-sup condition and show basic error estimates. Moreover we also show the stability of the fractional-step projection finite element scheme for the non-stationary problem. Finally, we give some numerical results to validate our method.     

Deadlock resolution using hand-to-hand motion

This paper deals with an approach for resolving deadlock problems in the case of carrying loads by the Distributed Autonomous Robotic System (DARS). The deadblock condition appears in multiple robots which move and work autonomously. Therefore, we propose an algorithm to resolve the deadlock condition by cooperative hand-to-hand motion, which is a type of cooperative performance. This paper shows the effectiveness of hand-to-hand motion through numerical simulation. This work was presented, in part, at International Symposium on Artificial Life and Robotics, Oita, Japan, February 18–20, 1996     

Improvement of Convergence to Steady State Solutions of Euler Equations with the WENO Schemes

The convergence to steady state solutions of the Euler equations for high order weighted essentially non-oscillatory (WENO) finite difference schemes with the Lax-Friedrichs flux splitting (Jiang and Shu, in J. Comput. Phys. 126:202–228, 1996) is investigated. Numerical evidence in Zhang and Shu (J. Sci. Comput. 31:273–305, 2007) indicates that there exist slight post-shock oscillations when we use high order WENO schemes to solve problems containing shock waves. Even though these oscillations are small in their magnitude and do not affect the “essentially non-oscillatory” property of the WENO schemes, they are indeed responsible for the numerical residue to hang at the truncation error level of the scheme instead of settling down to machine zero. Differently from the strategy adopted in Zhang and Shu (J. Sci. Comput. 31:273–305, 2007), in which a new smoothness indicator was introduced to facilitate convergence to steady states, in this paper we study the effect of the local characteristic decomposition on steady state convergence. Numerical tests indicate that the slight post-shock oscillation has a close relationship with the local characteristic decomposition process. When this process is based on an average Jacobian at the cell interface using the Roe average, as is the standard procedure for WENO schemes, such post-shock oscillation appears. If we instead use upwind-biased interpolation to approximate the physical variables including the velocity and enthalpy on the cell interface to compute the left and right eigenvectors of the Jacobian for the local characteristic decomposition, the slight post-shock oscillation can be removed or reduced significantly and the numerical residue settles down to lower values than other WENO schemes and can reach machine zero for many test cases. This new procedure is also effective for higher order WENO schemes and for WENO schemes with different smoothness indicators.     

Nonsmooth data error estimates for damped single step methods for parabolic equations in Banach space

We consider a time discretization method for a parabolic initial boundary value problem obtained from a combination of an A-stable single step method of order p and a lower order method with good smoothing properties. Such methods, including the Crank–Nicolson method combined with the backward Euler method, were analyzed in Hilbert space by Luskin and Rannacher, and nonsmooth data error estimates of order p were obtained. We extend this result to Banach space, and also consider approximations of the time derivative. Further, we apply the results to obtain error estimates in the supremum norm for fully discrete methods obtained by discretizing the space variable by a finite element method. Received: February 1998/ Accepted: November 1998