The coupled problem of a solid oscillating in a viscous fluid under the action of an elastic force

The torsional oscillations are studied of a solid of revolution under the action of elastic torque inside a container with a viscous incompressible fluid. We prove the asymptotic stability of the static equilibrium. We use the two approaches: the direct Lyapunov and linearization methods. The global asymptotic stability is established using a one-parameter family of Lyapunov functionals. Then small oscillations are studied of the fluid-solid system. The linearized operator of the problem of a solid oscillating in a fluid can be realized as an operator matrix obtained by appending two scalar rows and two columns to the Stokes operator. This operator is therefore a two-dimensional bordering of the Stokes operator and inherits many properties of the latter; in particular, the spectrum is discrete. The eigenvalue problem for the linearized operator is reduced to solving a dispersion equation. Inspection of the equation shows that all eigenvalues lie inside the right (stable) half-plane. Basing on this, we justify the linearization. Using an abstract theorem of Yudovich, we prove the asymptotic stability in a scale of function spaces, the infinite differentiability of solutions, and the decay of all their derivatives in time.     

Preconditioners for the p-version of the Galerkin method for a coupled finite element/boundary element system

We propose and analyze efficient preconditioners for solving systems of equations arising from the p-version for the finite element/boundary element coupling. The first preconditioner amounts to a block Jacobi method, whereas the second one is partly given by diagonal scaling. We use the generalized minimum residual method for the solution of the linear system. For our first preconditioner, the number of iterations of the GMRES necessary to obtain a given accuracy grows like log² p, where p is the polynomial degree of the ansatz functions. The second preconditioner, which is more easily implemented, leads to a number of iterations that behave like p log³ p. Computational results are presented to support this theory. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 47–61, 1998     

A variational conjugate gradient method for determining the fluid velocity of a slow viscous flow

The problem considered is that of determining the fluid velocity for linear hydrostatics Stokes flow of slow viscous fluids from measured velocity and fluid stress force on a part of the boundary of a bounded domain. A variational conjugate gradient iterative procedure is proposed based on solving a series of mixed well-posed boundary value problems for the Stokes operator and its adjoint. In order to stabilize the Cauchy problem, the iterations are ceased according to an optimal order discrepancy principle stopping criterion. Numerical results obtained using the boundary element method confirm that the procedure produces a convergent and stable numerical solution.     

A parallel version of the preconditioned conjugate gradient method for boundary element equations

The parallel version of precondition techniques is developed for matrices arising from the Galerkin boundary element method for two-dimensional domains with Dirichlet boundary conditions. Results were obtained for implementations on a transputer network as well as on an nCUBE-2 parallel computer showing that iterative solution methods are very well suited for a MIMD computer. A comparison of numerical results for iterative and direct solution methods is presented and underlines the superiority of iterative methods for large systems.     

On the coupling of regularization techniques and the boundary element method for a hemivariational inequality modelling a delamination problem

In this paper, we couple regularization techniques of nondifferentiable optimization with the h‐version of the boundary element method (h‐BEM) to solve nonsmooth variational problems arising in contact mechanics. As a model example, we consider the delamination problem. The variational formulation of this problem leads to a hemivariational inequality with a nonsmooth functional defined on the contact boundary. This problem is first regularized and then discretized by an h‐BEM. We prove convergence of the h‐BEM Galerkin solution of the regularized problem in the energy norm, provide an a priori error estimate and give a numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Completeness of the Floquet solutions to the problem of a solid oscillating in a fluid

We prove the completeness of the Floquet solutions to the parabolic equation describing small oscillations of a fluid-solid system. The symmetry axis of the solid is fixed inside a container of an arbitrary shape which is filled with an incompressible viscous fluid. The solid oscillates torsionally under the action of an elastic force with time periodic rigidity.     

A priori error estimates for a coupled finite element method and mixed finite element method for a fluid-solid interaction problem

This paper presents a heterogeneous finite element method fora fluid–solid interaction problem. The method, which combinesa standard finite element discretization in the fluid regionand a mixed finite element discretization in the solid region,allows the use of different meshes in fluid and solid regions.Both semi-discrete and fully discrete approximations are formulatedand analysed. Optimal order a priori error estimates in theenergy norm are shown. The main difficulty in the analysis iscaused by the two interface conditions which describe the interactionbetween the fluid and the solid. This is overcome by explicitlybuilding one of the interface conditions into the finite elementspaces. Iterative substructuring algorithms are also proposedfor effectively solving the discrete finite element equations.     

On the motion of rigid bodies in a viscous fluid

We consider the problem of motion of several rigid bodies in a viscous fluid. Both compressible and incompressible fluids are studied. In both cases, the existence of globally defined weak solutions is established regardless possible collisions of two or more rigid objects.     

Stability of a streamline diffusion finite element method for turning point problems

A one-dimensional singularly perturbed problem with a boundary turning point is considered in this paper. Let V^h be the linear finite element space on a suitable grid . A variant of streamline diffusion finite element method is proved to be almost uniform stable in the sense that the numerical approximation u_h satisfies u-u_h∞C|lnε| inf_vhV^hu-v_h∞, where C is independent with the small diffusion coefficient ε and the mesh . Such stability result is applied to layer-adapted grids to obtain almost ε-uniform second order scheme for turning point problems.     

On the convergence of the collocation method for nonlinear boundary integral equations

Recently, Galerkin and collocation methods have been analysed for some nonlinear boundary integral equations. For the collocation method it has been assumed that the nonlinearity is asymptotically linear. In this paper we remove this restriction. We shall prove the convergence of the collocation method for nonlinear boundary integral equations, when the nonlinearity has a polynomial growth condition. In addition to this the optimal order error estimates follow in L^q(Γ)-norm.     

Analysis of the stress singularity for a bi-material V-notch by the boundary element method

A new algorithm coupling the boundary element technique with the characteristic expansion method is proposed for the computation of the singular stress field in the V-notched bi-material structure. After the stress asymptotic expansions are introduced into the linear elasticity equilibrium equations, the governing equations at the small sector dug out from the bi-material V-notch tip region are transformed into the ordinary differential eigen-equations. All the parameters in the asymptotic expansions except the combination coefficients can be achieved by solving the established eigen-equations with the interpolating matrix method. Furthermore, the conventional boundary element method is applied to modeling the remaining structure without the notch tip region. The combination coefficients in the asymptotic expansion forms can be computed by the discretized boundary integral equations. Thus, the singular stress field at the V-notch tip and the generalized stress intensity factors of the bi-material notch are successfully calculated. The accurate singular stress field obtained here is very useful in the evaluation of the fracture property and the fatigue life of the V-notched bi-material structure.     

Efficiency of a posteriori BEM--error estimates for first-kind integral equations on quasi--uniform meshes

In the numerical treatment of integral equations of the first kind using boundary element methods (BEM), the author and E. P. Stephan have derived a posteriori error estimates as tools for both reliable computation and self-adaptive mesh refinement. So far, efficiency of those a posteriori error estimates has been indicated by numerical examples in model situations only. This work affirms efficiency by proving the reverse inequality. Based on best approximation, on inverse inequalities and on stability of the discretization, and complementary to our previous work, an abstract approach yields a converse estimate. This estimate proves efficiency of an a posteriori error estimate in the BEM on quasi--uniform meshes for Symm's integral equation, for a hypersingular equation, and for a transmission problem.

     

A nonlinear iteration method for solving a two-dimensional nonlinear coupled system of parabolic and hyperbolic equations

A nonlinear iteration method for solving a class of two-dimensional nonlinear coupled systems of parabolic and hyperbolic equations is studied. A simple iterative finite difference scheme is designed; the calculation complexity is reduced by decoupling the nonlinear system, and the precision is assured by timely evaluation updating. A strict theoretical analysis is carried out as regards the convergence and approximation properties of the iterative scheme, and the related stability and approximation properties of the nonlinear fully implicit finite difference (FIFD) scheme. The iterative algorithm has a linear constringent ratio; its solution gives a second-order spatial approximation and first-order temporal approximation to the real solution. The corresponding nonlinear FIFD scheme is stable and gives the same order of approximation. Numerical tests verify the results of the theoretical analysis. The discrete functional analysis and inductive hypothesis reasoning techniques used in this paper are helpful for overcoming difficulties arising from the nonlinearity and coupling and lead to a related theoretical analysis for nonlinear FI schemes.     

Peristaltic motion of a magnetohydrodynamic generalized second‐order fluid in an asymmetric channel

We investigate the peristaltic motion of a magnetohydrodynamic (MHD) generalized second‐order fluid in an asymmetric channel. The governing equations are first modeled and then numerically solved under the long wavelength approximation. Attention has been focused to analyze the shear‐thinning and shear‐thickening effects of the investigated non‐Newtonian fluid, the influence of the magnetic force on the flow, especially the trapping, pumping characteristics caused by the peristalsis of the walls. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Boundary element analysis of uncoupled transient thermo-elastic problems with time- and space-dependent heat sources

A boundary element method (BEM) for the analysis of two- and three-dimensional uncoupled transient thermo-elastic problems involving time- and space-dependent heat sources is presented. The domain integrals are efficiently treated using the Cartesian transformation and the radial integration methods without considering any internal cells. Similar to the dual reciprocity method (DRM), some internal points without any connectivity are considered; however, in contrast to the DRM, any arbitrary mesh-free interpolation method can be used in the present formulation. There is no need to find any particular solutions and the shape functions in the mesh-free interpolation method can be arbitrary and sufficiently complicated. Unlike the DRM, the generated system of equations contains the unknowns only on the boundary. After finding the primary unknowns on the boundary, the temperature, displacement, and stress components at all internal points can directly be found without solving any system of equations. Three examples with different forms of heat sources are presented to demonstrate the efficiency and accuracy of the proposed method. Although the proposed BEM is mathematically more complicated than domain methods, such as the finite element method (FEM), it is more efficient from a modelling viewpoint since only the surface mesh has to be generated in the presented method.     

Chaos in a restricted problem of rotation of a rigid body with a fixed point

In this paper, we consider the transition to chaos in the phase portrait of a restricted problem of rotation of a rigid body with a fixed point. Two interrelated mechanisms responsible for chaotization are indicated: (1) the growth of the homoclinic structure and (2) the development of cascades of period doubling bifurcations. On the zero level of the area integral, an adiabatic behavior of the system (as the energy tends to zero) is noted. Meander tori induced by the break of the torsion property of the mapping are found.      

A coupled BEM and FEM for the interior transmission problem in acoustics

The interior transmission problem (ITP) is a boundary value problem arising in inverse scattering theory, and it has important applications in qualitative methods. In this paper, we propose a coupled boundary element method (BEM) and a finite element method (FEM) for the ITP in two dimensions. The coupling procedure is realized by applying the direct boundary integral equation method to define the so-called Dirichlet-to-Neumann (DtN) mappings. We show the existence of the solution to the ITP for the anisotropic medium. Numerical results are provided to illustrate the accuracy of the coupling method.     

A boundary integral formulation for elastically deformable particles in a viscous fluid

This paper reports a formulation and implementation of a mixed (both direct and indirect) boundary element method using the double layer and its adjoint in a form suitable for solving Stokes flow problems involving elastically deformable particles. The formulation is essentially the Completed Double Layer Boundary Element Method for solving an exterior traction problem for the surrounding fluid or solid phase, followed by an interior displacement, and a mobility problem (if required) for the elastic particles. At the heart of the method is a deflation procedure that allows iterative solution strategies to be adopted, effectively opens the way for large-scale simulations of suspensions of deformable particles to be performed. Several problems are considered, to illustrate and benchmark the method. In particular, an analytical solution for an elastic sphere in an elongational flow is derived. The stresslet calculations for an elastic sphere in shear and elongational flows indicate that elasticity of the inclusions can potentially lead to positive second normal stress difference in shear flow, and an increase in the tensile resistance in elongational flow.This work is supported by a grant from the Australian Research Grant Council. X-J F wishes to acknowledge the support of the National Natural Science Foundation of China.     

Boundary elements and β-spline surface modeling for medical applications

The aim of this paper is to present and discuss an approach based on the integration of the boundary element method (BEM) with β-spline geometric modeling of the different surfaces involved in the external bone remodeling phenomena. The purpose of combining these two techniques is to avoid the jagged edges shapes and thus, to increase the convergence speed of the bone remodeling function. In this study, the external bone remodeling model proposed by Fridez et al. [P. Fridez, L. Rakotomanana, A. Terrier, P.F. Leyvraz, Three dimensional model of bone external adaptation, Comput. Methods Biomech. Biomed. Eng. 2 (1998) 189–196] is used. This model shows the change of the external bone surface remodeling at a boundary point, as a function of the stimulus variable Ψ. This variable is related to the stress tensor and the normal vector to that point. The β-spline surfaces were used because they are simple and reliable to smooth the contour by using the less possible number of geometric constraints. Some numerical examples are presented and discussed in order to show the versatility of the proposed approach.     

Conjugate gradient-boundary element solution using multiple reciprocity method for distributed elliptic optimal control problems

This paper deals with the numerical solution of optimal control problems, where the state equations are given by the fourth order elliptic partial differential equations. An iterative algorithm for this class of problems is developed. This new proposal is obtained by combining the Conjugate Gradient Method (CGM) with the Boundary Element Method (BEM) and Multiple Reciprocity Method (MRM). The local error estimates based on the stability of this scheme in the H² norm, L² norm and L^∞ norm are obtained. Finally, the numerical results on a test case show that this method is correct and feasible.