Time-harmonic solution for acousto-elastic interaction with controllability and spectral elements

The classical way of solving the time-harmonic linear acousto-elastic wave problem is to discretize the equations with finite elements or finite differences. This approach leads to large-scale indefinite complex-valued linear systems. For these kinds of systems, it is difficult to construct efficient iterative solution methods. That is why we use an alternative approach and solve the time-harmonic problem by controlling the solution of the corresponding time dependent wave equation.In this paper, we use an unsymmetric formulation, where fluid-structure interaction is modeled as a coupling between pressure and displacement. The coupled problem is discretized in space domain with spectral elements and in time domain with central finite differences. After discretization, exact controllability problem is reformulated as a least-squares problem, which is solved by the conjugate gradient method.     

Comparison between the shifted-Laplacian preconditioning and the controllability methods for computational acoustics

Processes that can be modelled with numerical calculations of acoustic pressure fields include medical and industrial ultrasound, echo sounding, and environmental noise. We present two methods for making these calculations based on Helmholtz equation. The first method is based directly on the complex-valued Helmholtz equation and an algebraic multigrid approximation of the discretized shifted-Laplacian operator; i.e. the damped Helmholtz operator as a preconditioner. The second approach returns to a transient wave equation, and finds the time-periodic solution using a controllability technique. We concentrate on acoustic problems, but our methods can be used for other types of Helmholtz problems as well. Numerical experiments show that the control method takes more CPU time, whereas the shifted-Laplacian method has larger memory requirement.     

Marching schemes for inverse acoustic scattering problems

Summary For the numerical solution of inverse Helmholtz problems the boundary value problem for a Helmholtz equation with spatially variable wave number has to be solved repeatedly. For large wave numbers this is a challenge. In the paper we reformulate the inverse problem as an initial value problem, and describe a marching scheme for the numerical computation that needs only n² log n operations on an n × n grid. We derive stability and error estimates for the marching scheme. We show that the marching solution is close to the low-pass filtered true solution. We present numerical examples that demonstrate the efficacy of the marching scheme.     

A meshfree numerical method for acoustic wave propagation problems in planar domains with corners and cracks

The numerical solution of acoustic wave propagation problems in planar domains with corners and cracks is considered. Since the exact solution of such problems is singular in the neighborhood of the geometric singularities the standard meshfree methods, based on global interpolation by analytic functions, show low accuracy. In order to circumvent this issue, a meshfree modification of the method of fundamental solutions is developed, where the approximation basis is enriched by an extra span of corner adapted non-smooth shape functions. The high accuracy of the new method is illustrated by solving several boundary value problems for the Helmholtz equation, modelling physical phenomena from the fields of room acoustics and acoustic resonance.     

On the super-approximation property of Galerkin's method with finite elements

Summary For Galerkin's method with finite elements as trial functions for strongly elliptic operator equations in the Hilbert scaleH ^t the super-approximation property and the optimal convergence rate are obtained by using the Aubin-Nitsche lemma. This applies in particular to spline collocation methods for a wide class of pseudodifferential equations.Dedicated to the memory of Professor Lothar Collatz     

Pseudospectral method for quadrilaterals

In this paper, we investigate the pseudospectral method on quadrilaterals. Some results on Legendre-Gauss-type interpolation are established, which play important roles in the pseudospectral method for partial differential equations defined on quadrilaterals. As examples of applications, we propose pseudospectral methods for two model problems and prove their spectral accuracy in space. Numerical results demonstrate the efficiency of the suggested algorithms. The approximation results and techniques developed in this paper are also applicable to other problems defined on quadrilaterals.     

A collocation method for high-frequency scattering by convex polygons

We consider the problem of scattering of a time-harmonic acoustic incident plane wave by a sound soft convex polygon. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave. Recently Chandler–Wilde and Langdon proposed a novel Galerkin boundary element method for this problem for which, by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space, they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency. Here we propose a related collocation method, using the same approximation space, for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the Galerkin scheme, but with a substantially reduced computational cost.     

Efficient evaluation of highly oscillatory acoustic scattering surface integrals

We consider the approximation of some highly oscillatory weakly singular surface integrals, arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three-dimensional convex obstacles, described globally in spherical coordinates. As the frequency of the incident wave increases, the performance of standard quadrature schemes deteriorates. Naive application of asymptotic schemes also fails due to the weak singularity. We propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system. For the case of a spherical scatterer we demonstrate via error analysis and numerical results that, provided the observation point is sufficiently far from the shadow boundary, a high level of accuracy can be achieved with a minimal computational cost.     

Analysis of one-dimensional Helmholtz equation with PML boundary

In this paper, the linear conforming finite element method for the one-dimensional Bérenger's PML boundary is investigated and well-posedness of the given equation is discussed. Furthermore, optimal error estimates and stability in the L²$L^2$ or H¹$H^1$-norm are derived under the assumption that h$h$, h²ω²$h^2{\omega }^2$ and h²ω³$h^2{\omega }^3$ are sufficiently small, where h$h$ is the mesh size and ω$\omega$ denotes a fixed frequency. Numerical examples are presented to validate the theoretical error bounds.     

Integration matrix based on arbitrary grids with a preconditioner for pseudospectral method

A matrix representation of integration for arbitrary grids is introduced. Suitable results are then obtained to be used along with differentiation matrix preconditioner to implement Pseudospectral method on integro-differential equations using arbitrary grids. Numerical examples are given to clarify the efficiency of the new method.     

Global extrapolation with a parallel splitting method

Extrapolation with a parallel splitting method is discussed. The parallel splitting method reduces a multidimensional problem into independent one-dimensional problems and can improve the convergence order of space variables to an order as high as the regularity of the solution permits. Therefore, in order to match the convergence order of the space variables, a high order method should also be used for the time integration. Second and third order extrapolation methods are used to improve the time convergence and it was found that the higher order extrapolation method can produce a more accurate solution than the lower order extrapolation method, but the convergence order of high order extrapolation may be less than the actual order of the extrapolation. We also try to show a fact that has not been studied in the literature, i.e. when the extrapolation is used, it may decrease the convergence of the space variables. The higher the order of the extrapolation method, the more it decreases the convergence of the space variables. The global extrapolation method also improves the parallel degree of the parallel splitting method. Numerical tests in the paper are done in a domain of a unit circle and a unit square.Supported by the Academy of Finland.     

Solution of the Dirichlet and Neumann problems for a modified Helmholtz equation in Besov spaces on an annulus

Here we study Dirichlet and Neumann problems for a special Helmholtz equation on an annulus. Our main aim is to measure smoothness of solutions for the boundary datum in Besov spaces. We shall use operator theory to solve this problem. The most important advantage of this technique is that it enables to consider equations in vector-valued settings. It is interesting to note that optimal regularity of this problem will be a special case of our main result.     

Linear and bilinear immersed finite elements for planar elasticity interface problems

This article is to discuss the linear (which was proposed in  and ) and bilinear immersed finite element (IFE) methods for solving planar elasticity interface problems with structured Cartesian meshes. Basic features of linear and bilinear IFE functions, including the unisolvent property, will be discussed. While both methods have comparable accuracy, the bilinear IFE method requires less time for assembling its algebraic system. Our analysis further indicates that the bilinear IFE functions are guaranteed to be applicable to a larger class of elasticity interface problems than linear IFE functions. Numerical examples are provided to demonstrate that both linear and bilinear IFE spaces have the optimal approximation capability, and that numerical solutions produced by a Galerkin method with these IFE functions for elasticity interface problem also converge optimally in both L²$L^2$ and semi-H¹$H^1$ norms.     

An ultra-weak method for acoustic fluid–solid interaction

We introduce the ultra-weak variational formulation (UWVF) for fluid–solid vibration problems. In particular, we consider the scattering of time-harmonic acoustic pressure waves from solid, elastic objects. The problem is modeled using a coupled system of the Helmholtz and Navier equations. The transmission conditions on the fluid–solid interface are represented in an impedance-type form after which we can employ the well known ultra-weak formulations for the Helmholtz and Navier equations. The UWVF approximation for both equations is computed using a superposition of propagating plane waves. A condition number based criterion is used to define the plane wave basis dimension for each element. As a model problem we investigate the scattering of sound from an infinite elastic cylinder immersed in a fluid. A comparison of the UWVF approximation with the analytical solution shows that the method provides a means for solving wave problems on relatively coarse meshes. However, particular care is needed when the method is used for problems at frequencies near the resonance frequencies of the fluid–solid system.     

A direct identification method for current dipoles from electromagnetic fields

We discuss the identification problem for current dipoles in a spherically symmetric conductor. This mathematical model is used for a biomedical inverse problem such as the source current identification for the human brain activity. We have already proposed a direct identification method for this inverse source problem using observations of the magnetic fields outside of the conductor. One of the difficulties of current dipole identification using the magnetic fields is caused by the fact that magnetic field does not include any information about the radial component of dipole moments. In this paper, we consider an improvement of the direct method to identify both radial and tangential components of current dipole moments by combining electric and magnetic observation data. Furthermore, our approach is effective in the case where the number of dipoles is unknown.     

Two-scale composite finite element method for Dirichlet problems on complicated domains

In this paper, we define a new class of finite elements for the discretization of problems with Dirichlet boundary conditions. In contrast to standard finite elements, the minimal dimension of the approximation space is independent of the domain geometry and this is especially advantageous for problems on domains with complicated micro-structures. For the proposed finite element method we prove the optimal-order approximation (up to logarithmic terms) and convergence estimates valid also in the cases when the exact solution has a reduced regularity due to re-entering corners of the domain boundary. Numerical experiments confirm the theoretical results and show the potential of our proposed method.     

Carleman estimates for degenerate parabolic operators with applications to null controllability

We prove an estimate of Carleman type for the one dimensional heat equation $$ u_t - \left( {a\left( x \right)u_x } \right)_x + c\left( {t,x} \right)u = h\left( {t,x} \right),\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where a(·) is degenerate at 0. Such an estimate is derived for a special pseudo-convex weight function related to the degeneracy rate of a(·). Then, we study the null controllability on [0, 1] of the semilinear degenerate parabolic equation $$ u_t - \left( {a\left( x \right)u_x } \right)_x + f\left( {t,x,u} \right) = h\left( {t,x} \right)\chi _\omega \left( x \right), $$ where (t, x) ∈(0, T) × (0, 1), ω=(α, β) ⊂⊂ [0, 1], and f is locally Lipschitz with respect to u. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in R2

Summary In this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior Dirichlet problem in the plane. Specifically, the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region, and the Laplace equation in the corresponding unbounded exterior region, in addition to appropriate boundary and transmission conditions. The main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation. We provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence, uniqueness and approximation results are established. Then, we consider the case where the corresponding operator is strongly monotone and Lipschitz-continuous, and derive asymptotic error estimates for a boundary-finite element solution. We prove the unique solvability of the discrete operator equations, and based on a Strang type abstract error estimate, we show the strong convergence of the approximated solutions. Moreover, under additional regularity assumptions on the solution of the continous operator equation, the asymptotic rate of convergenceO (h) is obtained.The first author's research was partly supported by the U.S. Army Research Office through the Mathematical Science Institute of Cornell University, by the Universidad de Concepción through the Facultad de Ciencias, Dirección de Investigación and Vicerretoria, and by FONDECYT-Chile through Project 91-386.     

Convergence estimates for an higher order optimized Schwarz method for domains with an arbitrary interface

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. Optimized Schwarz methods employ a first or higher order boundary condition along the artificial interface to accelerate convergence. In the literature, the analysis of optimized Schwarz methods relies on Fourier analysis and so the domains are restricted to be regular (rectangular). In this paper, we express the interface operator of an optimized Schwarz method in terms of Poincare-Steklov operators. This enables us to derive an upper bound of the spectral radius of the operator arising in this method of 1−O(h^1/4) on a class of general domains, where h is the discretization parameter. This is the predicted rate for a second order optimized Schwarz method in the literature on rectangular subdomains and is also the observed rate in numerical simulations.