An official website of the United States government
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
Weighted integral solvers for elastic scattering by open arcs in two dimensions
Abstract We present new methodologies for the numerical solution of problems of elastic scattering by open arcs in two dimensions. The algorithms utilize weighted versions of the classical elastic integral operators associated with Dirichlet and Neumann boundary conditions, where the integral weight accounts for (and regularizes) the singularity of the integral‐equation solutions at the open‐arc endpoints. Crucially, the method also incorporates a certain “open‐arc elastic Calderón relation” introduced in this article, whose validity is demonstrated on the basis of numerical experiments, but whose rigorous mathematical proof is left for future work. (In fact, the aforementioned open‐arc elastic Calderón relation generalizes a corresponding elastic Calderón relation for closed surfaces, which is also introduced in this article, and for which a rigorous proof is included.) Using the open‐surface Calderón relation in conjunction with spectrally accurate quadrature rules and the Krylov‐subspace linear algebra solver GMRES, the proposed overall open‐arc elastic solver produces results of high accuracy in small number of iterations, for both low and high frequencies. A variety of numerical examples in this article demonstrate the accuracy and efficiency of the proposed methodology.
more »
« less
- Award ID(s):
- 1714169
- PAR ID:
- 10452536
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- International Journal for Numerical Methods in Engineering
- Volume:
- 122
- Issue:
- 11
- ISSN:
- 0029-5981
- Page Range / eLocation ID:
- p. 2733-2750
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Brenner, Susan (Ed.)This paper proposes a frequency-time hybrid solver for the time-dependent wave equation in two-dimensionalinterior spatial domains. The approach relies on four main elements, namely, (1) A multiple scattering strategy that decomposes a giveninteriortime-domain problem into a sequence oflimited-durationtime-domain problems of scattering by overlapping open arcs, each one of which is reduced (by means of the Fourier transform) to a sequence ofHelmholtz frequency-domain problems; (2) Boundary integral equations on overlapping boundary patches for the solution of the frequency-domain problems in point (1); (3) A smooth“Time-windowing and recentering”methodology that enables both treatment of incident signals of long duration and long time simulation; and, (4) A Fourier transform algorithm that delivers numerically dispersionless,spectrally-accurate time evolutionfor given incident fields. By recasting the interior time-domain problem in terms of a sequence of open-arc multiple scattering events, the proposed approach regularizes the full interior frequency domain problem—which, if obtained by either Fourier or Laplace transformation of the corresponding interior time-domain problem, must encapsulate infinitely many scattering events, giving rise to non-uniqueness and eigenfunctions in the Fourier case, and ill conditioning in the Laplace case. Numerical examples are included which demonstrate the accuracy and efficiency of the proposed methodology.more » « less
-
null (Ed.)Abstract The matrix elements of relativistic nucleon–nucleon ( NN ) potentials are calculated directly from the nonrelativistic potentials as a function of relative NN momentum vectors, without a partial wave decomposition. To this aim, the quadratic operator relation between the relativistic and nonrelativistic NN potentials is formulated in momentum-helicity basis states. It leads to a single integral equation for the two-nucleon (2 N ) spin-singlet state, and four coupled integral equations for two-nucleon spin-triplet states, which are solved by an iterative method. Our numerical analysis indicates that the relativistic NN potential obtained using CD-Bonn potential reproduces the deuteron binding energy and neutron-proton elastic scattering differential and total cross-sections with high accuracy.more » « less
-
Bruno, Oscar; Pandey, Ambuj (Ed.)This paper presents a fast high-order method for the solution of two-dimensional problems of scattering by penetrable inhomogeneous media, with application to high-frequency configurations containing (possibly) discontinuous refractivities. The method relies on a hybrid direct/iterative combination of 1)~A differential volumetric formulation (which is based on the use of appropriate Chebyshev differentiation matrices enacting the Laplace operator) and, 2)~A second-kind boundary integral formulation (which, once again, utilizes Chebyshev discretization, but, in this case, in the boundary-integral context). The approach enjoys low dispersion and high-order accuracy for smooth refractivities, as well as second-order accuracy (while maintaining low dispersion) in the discontinuous refractivity case. The solution approach proceeds by application of Impedance-to-Impedance (ItI) maps to couple the volumetric and boundary discretizations. The volumetric linear algebra solutions are obtained by means of a multifrontal solver, and the coupling with the boundary integral formulation is achieved via an application of the iterative linear-algebra solver GMRES. In particular, the existence and uniqueness theory presented in the present paper provides an affirmative answer to an open question concerning the existence of a uniquely solvable second-kind ItI-based formulation for the overall scattering problem under consideration. Relying on a modestly-demanding scatterer-dependent precomputation stage (requiring in practice a computing cost of the order of $$O(N^{\alpha})$$ operations, with $$\alpha \approx 1.07$$, for an $$N$$-point discretization \textcolor{black}{and for the relevant Chebyshev accuracy orders $$q$$ used)}, together with fast ($O(N)$-cost) single-core runs for each incident field considered, the proposed algorithm can effectively solve scattering problems for large and complex objects possibly containing discontinuities and strong refractivity contrasts.more » « less
-
Natural dynamics, nonlinear optimization, and, more recently, convex optimization are available methods for stiffness design of energy-efficient series elastic actuators. Natural dynamics and general nonlinear optimization only work for a limited set of load kinetics and kinematics, cannot guarantee convergence to a global optimum, or depend on initial conditions to the numerical solver. Convex programs alleviate these limitations and allow a global solution in polynomial time, which is useful when the space of optimization variables grows (e.g., when designing optimal nonlinear springs or co-designing spring, controller, and reference trajectories). Our previous work introduced the stiffness design of series elastic actuators via convex optimization when the transmission dynamics are negligible, which is an assumption that applies mostly in theory or when the actuator uses a direct or quasi-direct drive. In this work, we extend our analysis to include friction at the transmission. Coulomb friction at the transmission results in a non-convex expression for the energy dissipated as heat, but we illustrate a convex approximation for stiffness design. We experimentally validated our framework using a series elastic actuator with specifications similar to the knee joint of the Open Source Leg, an open-source robotic knee-ankle prosthesis.more » « less
