Consider the scattering of a time-harmonic acoustic plane wave by a bounded elastic obstacle which is immersed in a homogeneous acoustic medium. This paper is concerned with an inverse acoustic-elastic interaction problem, which is to determine the location and shape of the elastic obstacle by using either the phased or phaseless far-field data. By introducing the Helmholtz decomposition, the model problem is reduced to a coupled boundary value problem of the Helmholtz equations. The jump relations are studied for the second derivatives of the single-layer potential in order to deduce the corresponding boundary integral equations. The well-posedness is discussed for the solution of the coupled boundary integral equations. An efficient and high order Nyström-type discretization method is proposed for the integral system. A numerical method of nonlinear integral equations is developed for the inverse problem. For the case of phaseless data, we show that the modulus of the far-field pattern is invariant under a translation of the obstacle. To break the translation invariance, an elastic reference ball technique is introduced. We prove that the inverse problem with phaseless far-field pattern has a unique solution under certain conditions. In addition, a numerical method of the reference ball technique based nonlinear integral equations is proposed for the phaseless inverse problem. Numerical experiments are presented to demonstrate the effectiveness and robustness of the proposed methods.
more »
« less
A problem in control of elastodynamics with piezoelectric effects
We consider an optimal control problem where the state equations are a coupled hyperbolic–elliptic system. This system arises in elastodynamics with piezoelectric effects—the elastic stress tensor is a function of elastic displacement and electric potential. The electric flux acts as the control variable and bound constraints on the control are considered. We develop a complete analysis for the state equations and the control problem. The requisite regularity on the control, to show the well-posedness of the state equations, is enforced using the cost functional. We rigorously derive the first-order necessary and sufficient conditions using adjoint equations and further study their well-posedness. For spatially discrete (time-continuous) problems, we show the convergence of our numerical scheme. Three-dimensional numerical experiments are provided showing convergence properties of a fully discrete method and the practical applicability of our approach.
more » « less- Award ID(s):
- 1818772
- NSF-PAR ID:
- 10128244
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- IMA Journal of Numerical Analysis
- ISSN:
- 0272-4979
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Consider the elastic scattering of a time-harmonic wave by multiple well-separated rigid particles with smooth boundaries in two dimensions. Instead of using the complex Green's tensor of the elastic wave equation, we utilize the Helmholtz decomposition to convert the boundary value problem of the elastic wave equation into a coupled boundary value problem of the Helmholtz equation. Based on single, double, and combined layer potentials with the simpler Green's function of the Helmholtz equation, we present three different boundary integral equations for the coupled boundary value problem. The well-posedness of the new integral equations is established. Computationally, a scattering matrix based method is proposed to evaluate the elastic wave for arbitrarily shaped particles. The method uses the local expansion for the incident wave and the multipole expansion for the scattered wave. The linear system of algebraic equations is solved by GMRES with fast multipole method (FMM) acceleration. Numerical results show that the method is fast and highly accurate for solving elastic scattering problems with multiple particles.more » « less
-
Granular flows occur in a wide range of situations of practical interest to industry, in our natural environment and in our everyday lives. This paper focuses on granular flow in the so-called inertial regime, when the rheology is independent of the very large particle stiffness. Such flows have been modelled with the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -rheology, which postulates that the bulk friction coefficient $\unicode[STIX]{x1D707}$ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction $\unicode[STIX]{x1D719}$ are functions of the inertial number $I$ only. Although the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -rheology has been validated in steady state against both experiments and discrete particle simulations in several different geometries, it has recently been shown that this theory is mathematically ill-posed in time-dependent problems. As a direct result, computations using this rheology may blow up exponentially, with a growth rate that tends to infinity as the discretization length tends to zero, as explicitly demonstrated in this paper for the first time. Such catastrophic instability due to ill-posedness is a common issue when developing new mathematical models and implies that either some important physics is missing or the model has not been properly formulated. In this paper an alternative to the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -rheology that does not suffer from such defects is proposed. In the framework of compressible $I$ -dependent rheology (CIDR), new constitutive laws for the inertial regime are introduced; these match the well-established $\unicode[STIX]{x1D707}(I)$ and $\unicode[STIX]{x1D6F7}(I)$ relations in the steady-state limit and at the same time are well-posed for all deformations and all packing densities. Time-dependent numerical solutions of the resultant equations are performed to demonstrate that the new inertial CIDR model leads to numerical convergence towards physically realistic solutions that are supported by discrete element method simulations.more » « less
-
In this paper we introduce a constructive approach to study well-posedness of solutions to stochastic uid-structure interaction with stochastic noise. We focus on a benchmark problem in stochastic uidstructure interaction, and prove the existence of a unique weak solution in the probabilistically strong sense. The benchmark problem consists of the 2D time-dependent Stokes equations describing the ow of an incompressible, viscous uid interacting with a linearly elastic membrane modeled by the 1D linear wave equation. The membrane is stochastically forced by the time-dependent white noise. The uid and the structure are linearly coupled. The constructive existence proof is based on a time-discretization via an operator splitting approach. This introduces a sequence of approximate solutions, which are random variables. We show the existence of a subsequence of approximate solutions which converges, almost surely, to a weak solution in the probabilistically strong sense. The proof is based on uniform energy estimates in terms of the expectation of the energy norms, which are the backbone for a weak compactness argument giving rise to a weakly convergent subsequence of probability measures associated with the approximate solutions. Probabilistic techniques based on the Skorohod representation theorem and the Gyongy-Krylov lemma are then employed to obtain almost sure convergence of a subsequence of the random approximate solutions to a weak solution in the probabilistically strong sense. The result shows that the deterministic benchmark FSI model is robust to stochastic noise, even in the presence of rough white noise in time. To the best of our knowledge, this is the rst well-posedness result for stochastic uid-structure interaction.more » « less
-
We consider an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with an $ L^2 $ penalty on the boundary control. The contribution of this paper is twofold. First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain. The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary. Second, we propose and analyze a hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For convex polygonal domains, our theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary. We present numerical experiments to demonstrate the performance of the HDG method.more » « less
