Convergence of a lowest-order finite element method for the transmission eigenvalue problem

The transmission eigenvalue problem arises in scattering theory. The main difficulty in its analysis is the fact that, depending on the chosen formulation, it leads either to a quadratic eigenvalue problem or to a non-classical mixed problem. In this paper we prove the convergence of a mixed finite element approximation. This approach, which is close to the Ciarlet–Raviart discretization of biharmonic problems, is based on Lagrange finite elements and is one of the less expensive methods in terms of the amount of degrees of freedom. The convergence analysis is based on classical abstract spectral approximation result and the theory of mixed finite element methods for solving the stream function–vorticity formulation of the Stokes problem. Numerical experiments are reported in order to assess the efficiency of the method.     

Chebyshev pseudospectral method of viscous flows with corner singularities

Chebyshev pseudospectral solutions of the biharmonic equation governing two-dimensional Stokes flow within a driven cavity converge poorly in the presence of corner singularities. Subtracting the strongest corner singularity greatly improves the rate of convergence. Compared to the usual stream function/ vorticity formulation, the single fourth-order equation for stream function used here has half the number of coefficients for equivalent spatial resolution and uses a simpler treatment of the boundary conditions. We extend these techniques to small and moderate Reynolds numbers.     

A priori and a posteriori error analysis for a large-scale ocean circulation finite element model

We consider the finite element solution of the stream function–vorticity formulation for a large-scale ocean circulation model. First, we study existence and uniqueness of solution for the continuous and discrete problems. Under appropriate regularity assumptions we prove that the stream function can be computed with an error of order h in H¹-seminorm. Second, we introduce and analyze an h-adaptive mesh refinement strategy to reduce the spurious oscillations and poor resolution which arise when convective terms are dominant. We propose an a posteriori anisotropic error indicator based on the recovery of the Hessian from the finite element solution, which allows us to obtain well adapted meshes. The numerical experiments show an optimal order of convergence of the adaptive scheme. Furthermore, this strategy is efficient to eliminate the oscillations around the boundary layer.     

Fluid-structure interactions using different mesh motion techniques

In this work, we compare different mesh moving techniques for monolithically-coupled fluid-structure interactions in arbitrary Lagrangian–Eulerian coordinates. The mesh movement is realized by solving an additional partial differential equation of harmonic, linear-elastic, or biharmonic type. We examine an implementation of time discretization that is designed with finite differences. Spatial discretization is based on a Galerkin finite element method. To solve the resulting discrete nonlinear systems, a Newton method with exact Jacobian matrix is used. Our results show that the biharmonic model produces the smoothest meshes but has increased computational cost compared to the other two approaches.     

A numerical solution of the Navier-Stokes equations using the finite element technique

The finite element discretisation technique is used to effect a solution of the Navier- Stokes equations. Two methods of formulation are presented, and a comparison of the effeciency of the methods, associated with the solution of particular problems, is made. The first uses velocity and pressure as field variables and the second stream function and vorticity. It appears that, for contained flow problems the first formulation has some advantages over previous approaches using the finite elemental method[1,2].     

Viscous flow past circular cylinders

Accurate and efficient calculation of the flow around a circular cylinder are presented for Reynolds number from 1 to 40 (based on the diameter). The semi-analytical method of series truncation is used to express the stream function and the vorticity in a finite Fourier series. Substitution into the Navier-Stokes equation yields a finite system of nonlinear ordinary differential equations. These are approximated by two-point boundary value methods. The resulting nonlinear equation are solved unsing Newton's method. Much attention has been given to numerical errors and errors resulting from the approximation by a finite series. The results are compared with similar calculations by Keller-Takami and Dennis-Chang. The agreement is good. The application of free stream conditions at a finite radius is shown to yield inaccurate results. im     

Mathematical aspect of the stream-function equations of compressible turbomachinery flows and their finite element approximations using optimal control

In this paper we establish the partial differential equation and properly pose conditions for the stream function on an arbitrary stream surface in a turbomachine using tensor analysis and a semi-geodesic coordinate system. We provide computational examples using the finite element method and also the error estimate of Galerkin's finite element approximation which depends on the Mach number.     

Numerical study of a dual iterative method for solving a finite element approximation of the biharmonic equation

The purpose of this paper is a presentation of a numerical study of an iterative method due to Ciarlet and Glowinski for solving a finite element approximation of the Dirichlet problem for the biharmonic operator. The main feature of this method is that it reduces the biharmonic problem to a sequence of Dirichlet problems for the operator -Δ. Therefore, in numerical examples, finite element programs for solving second-order problems can be used, and this is an interesting feature of the method.     

Solving biharmonic equation using the localized method of approximate particular solutions

Some localized numerical methods, such as finite element and finite difference methods (FDMs), have encountered difficulties when solving fourth or higher order differential equations. Localized methods, which use radial basis functions, are considered the generalized FDMs and, thus, inherit the similar difficulties when solving higher order differential equations. In this paper, we deal with the use of the localized method of approximate particular solutions (LMAPS), a recently developed localized radial basis function collocation method, in solving two-dimensional biharmonic equation in a bounded region. The technique is based on decoupling the biharmonic problem into two Poisson equations, and then the LMAPS is applied to each Poisson's problem to compute numerical solutions. Furthermore, the influence of the shape parameter and different radial basis functions on the numerical solution is discussed. The effectiveness of the proposed method is demonstrated by solving three examples in both regular and irregular domains.     

Spectral method of decoupling the vorticity and stream function for the incompressible fluid flows

A spectral method is proposed for the vorticity-stream function equations of the incompressible fluid flows. It is effective to overcome the lack of vorticity boundary condition. This method decouples the vorticity and stream function. At each time step, first, the vorticity is explicitly solved and the stream function is evaluated by a Poisson-like equation; then the vorticity is determined by a Poisson-like equation again. The numerical experiments show that this method is of efficiency and high accuracy.     

The importance of the exact satisfaction of the incompressibility constraint in nonlinear elasticity: mixed FEMs versus NURBS-based approximations

In the present paper we investigate the capability of finite element methods to correctly reproduce the stability range of finite strain problems in the incompressible regime. To this end, we develop a numerical scheme, obtained combining a stream function formulation with an isogeometric NURBS approach, which is able to sharply estimate the stability limits of the continuous problem. Using such a method, we show a pair of benchmark problems on which various well-known finite element methods largely fail in approximating the correct stability range.     

Approximate polynomial preconditionings applied to biharmonic equations

Applying a finite difference approximation to a biharmonic equation results in a very ill conditioned system of equations. This paper examines the conjugate gradient method used with polynomial preconditioning techniques for solving such linear systems. A new approach using an approximate polynomial preconditioner is described. The preconditioner is constructed from a series approximation based on the Laplacian finite difference matrix. A particularly attractive feature of this approach is that the Laplacian matrix consists of far fewer non-zero entries than the biharmonic finite difference matrix. Moreover, analytical estimates and computational results show that this preconditioner is more effective (in terms of the rate of convergence and the computational work required per iteration) than the polynomial preconditioner based on the original biharmonic matrix operator. The conjugate gradient algorithm and the preconditioning step can be efficiently implemented on a vector super-computer such as the CDC CYBER 205.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada Grant U0375; and in part by NASA (funded under the Space Act Agreement C99066G) while the author was visiting ICOMP, NASA Lewis Research Center.The work of this author was supported by an Izaak Walton Killam Memorial Scholarship.     

Numerical studies of the flow around a circular cylinder by a finite element method

Numerical solutions of the steady, incompressible, viscous flow past a circular cylinder are presented for Reynolds numbers R ranging from 1 to 100. The governing Navier-Stokes equations in the form of a single, fourth order differential equation for stream function and the boundary conditions are replaced by an equivalent variational principle. The numerical method is based on a finite element approximation of this principle. The resulting non-linear system is solved by the Newton-Raphson process. The pressure field is obtained from a finite element solution of the Poisson equation once the stream function is known. The results are compared with those determined by other numerical techniques and experiments. In particular, the discussion is concerned with the development of the closed wake with Reynolds number, and the tendency of R ≥ 40 flow toward instability.     

Development of a meshless Galerkin boundary node method for viscous fluid flows

In this paper, a meshless Galerkin boundary node method is developed for boundary-only analysis of the interior and exterior incompressible viscous fluid flows, governed by the Stokes equations, in biharmonic stream function formulation. This method combines scattered points and boundary integral equations. Some of the novel features of this meshless scheme are boundary conditions can be enforced directly and easily despite the meshless shape functions lack the delta function property, and system matrices are symmetric and positive definite. The error analysis and convergence study of both velocity and pressure are presented in Sobolev spaces. The performance of this approach is illustrated and assessed through some numerical examples.     

Structural optimization using global stress-deviation objective function via the level-set method

The paper deals with minimum stress design using a novel stress-related objective function based on the global stress-deviation measure. The shape derivative, representing the shape sensitivity analysis of the structure domain, is determined for the generalized form of the global stress-related objective function. The optimization procedure is based on the domain boundary evolution via the level-set method. The elasticity equations are, instead of using the usual ersatz material approach, solved by the extended finite element method. The Hamilton-Jacobi equation is solved using the streamline diffusion finite element method. The use of finite element based methods allows a unified numerical approach with only one numerical framework for the mechanical problem as also for the boundary evolution stage. The numerical examples for the L-beam benchmark and the notched beam are given. The results of the structural optimization problem, in terms of maximum von Mises stress corresponding to the obtained optimal shapes, are compared for the commonly used global stress measure and the novel global stress-deviation measure, used as the stress-related objective functions.     

High order finite difference methods for unsteady incompressible flows in multi-connected domains

Using the vorticity and stream function variables is an effective way to compute 2-D incompressible flow due to the facts that the incompressibility constraint for the velocity is automatically satisfied, the pressure variable is eliminated, and high order schemes can be efficiently implemented. However, a difficulty arises in a multi-connected computational domain in determining the constants for the stream function on the boundary of the “holes”. This is an especially challenging task for the calculation of unsteady flows, since these constants vary with time to reflect the total fluxes of the flow in each sub-channel. In this paper, we propose an efficient method in a finite difference setting to solve this problem and present some numerical experiments, including an accuracy check of a Taylor vortex-type flow, flow past a non-symmetric square, and flow in a heat exchanger.     

Design multiple-layer gradient coils using least-squares finite element method

The design of gradient coils for magnetic resonance imaging is an optimization task in which a specified distribution of the magnetic field inside a region of interest is generated by choosing an optimal distribution of a current density geometrically restricted to specified non-intersecting design surfaces, thereby defining the preferred coil conductor shapes. Instead of boundary integral type methods, which are widely used to design coils, this paper proposes an optimization method for designing multiple layer gradient coils based on a finite element discretization. The topology of the gradient coil is expressed by a scalar stream function. The distribution of the magnetic field inside the computational domain is calculated using the least-squares finite element method. The first-order sensitivity of the objective function is calculated using an adjoint equation method. The numerical operations needed, in order to obtain an effective optimization procedure, are discussed in detail. In order to illustrate the benefit of the proposed optimization method, example gradient coils located on multiple surfaces are computed and characterised.     

Dual iterative techniques for solving a finite element approximation of the biharmonic equation

A finite element approximation of the Dirichlet problem for the biharmonic operator is described. Its main feature is that it is equivalent to solving a sequence of discrete Dirichlet problems for the operator -Δ. This method, which has already been shown to be convergent, is particularly well-suited for problems in fluid dynamics.     

Numerical integration of the time-dependent equations of motion for taylor vortex flow

A method is presented for the finite difference solution of the equations of fluid motion. The complete Navier-Stokes equations are expressed in terms of tangential velocity, vorticity and stream function. The transformed equations are solved using an alternating direction implicit scheme. The classical problem of hydrodynamic stability of the rotational Couette flow is solved in two dimensions. Comparison with other numerical and experimental works shows that the method reported here is computationally stable, even when used with coarse grids and relatively large time increments.     

Rotational fluid flow using a least squares collocation technique

A potential theory approach for incompressible viscous flow which leads to the biharmonic equation is first developed. A numerical least squares collocation technique using fundamental singular solutions of the biharmonic equation is then applied to a rotational flow problem with moving boundaries that produce discontinuous boundary conditions associated with the biharmonic. It is shown that the least squares technique smoothes out local disturbances in boundary data of the type which are likely to present difficulties to the more commonly used boundary element method. A compact computer program for the method and the results for the problem of a rectangular channel with one moving boundary are included along with an experimental verification of the results using the thin plate bending analogy.