{tonatiuh@cims.nyu.edu} http://cims.nyu.edu/~tonatiuh/ I am especially interested in devising, rigorously analyzing and implementing methods for the numerical simulation of transient wave phenomena. During my doctoral studies I have focused on Boundary Integral Equation techniques for the simulation of time domain linear wave scattering and transmission problems. Prior to that I was involved in the simulation of nonlinear wave propagation by shock-capturing Godunov-style methods. Currently my main interest is the study and computational simulation of elastic wave propagation and its interaction with other kinds of waves and physical processes (acoustic, electric, poroelastic, fluid flow, etc.). I work in the design and analysis of numerical methods for their simulation using for instance boundary element, continuous and discontinuous Galerkin, or finite volume methods and the potential application of these techniques to direct and inverse problems of applied sciences and engineering. Boundary Integral Equations in the Time Domain and Convolution Quadrature Consider a linear time-independent elliptic partial differential operator L with an associated fundamental solution G(x,y). The solution u to the partial differential equation Lu = 0 in Ω R d, satisfying appropriate boundary conditions on Ω (and radiation conditions as x if Ω is unbounded) admits an integral representation in terms of the fundamental solution of the form u(x) = G(x,y)λ(y)dy ν(y) G(x,y)ϕ(y)dy, Ω where λ and ϕ are unknown density functions defined on Ω. The boundary conditions of the problem are then used to set up integral equations on the boundary from which the density functions can be retrieved. Boundary integral formulations exploit the possibility to transfer the volume problem onto the boundary of the region, effectively reducing the dimensionality of the problem by one. Moreover, once the equations have been discretized, all computations are done solely on the boundary, making these methods particularly well suited for the treatment of exterior problems in unbounded domains with homogeneous physical properties. In the time domain an analogous procedure can be followed resulting in boundary integral equations that have a convolutional structure. For instance, in R 3, the single layer potential for the wave equation has the following expression (S λ)(x,t) := Γ Ω λ(y,t c 1 x y ) dγ(y). 4π x y In this case, Convolution Quadrature [11, 12] (CQ) constitutes a very powerful tool for discretization. It relies on the observation that the convolution of a causal function g with a causal Laplace-transformable kernel f can be written in terms of its Laplace transform, F, and the solution to a first order evolution equation whose inhomogeneous term is exactly g. Formally, for some σ > 0 (f g)(t) = 1 2πi σ+i t σ i where y(t;s) is the unique solution of 0 F(s)e s(t τ) g(τ)dτ ds = 1 2πi ẏ(t) sy(t) = g(t), y(0) = 0. σ+i σ i F(s)y(t;s)ds, This differential equation can be approximately solved by any ODE-solving procedure using time-domain information from g. The numerical solution can then be used to approximate the complex contour integral using the generally more tractable Laplace-domain kernel F. Currently it is standard to use low order backward differentiation, trapezoidal rule or Runge-Kutta based CQ implementations for which 1
the behaviour of the error and convergence properties are well understood. In the near future I plan to explore the posibility of developing higher order schemes involving deferred correction to attain enhanced performance in applications of CQ to wave propagation. Traditionally, the stability and convergence analysis of the solutions obtained following this procedure depended on carefully bounding the operators of the Calderón projector associated to the PDE in terms of the Laplace parameter s and then applying results related to the inversion of the Laplace transform. In [8] we use a novel approach that sidesteps the Laplace-domain analysis using the machinery of abstract evolution equations and semi-group theory to obtain time-domain estimates and regularity results directly for the equations of acoustics. For wave-structure interaction, the presence of time derivatives in the coupling conditions adds difficulty to the problem. Nevertheless, we are currently working on the extension of these analysis techniques to such problems, in particular to the acoustic/elastic and acoustic/piezoelectric interaction [2]. Discretization of Boundary Integral Equations with reduced quadrature (deltabem) For smooth parametrizable scatterers in 2D we devised and implemented an inexpensive third-order method based on an improvement of ǫ-quadrature, a technique dating back to the work of Saranen and Vainikko in the 90 s [15]. The method, which we refer to as deltabem, is built on the idea of collocating the equation on equispaced points in parametric space and then approximating the integral operators using a one-point quadrature rule in carefully shifted symmetric grids as shown in Figure 1. The technique can be understood as a non-conforming Petrov-Galerkin discretization using piecewise constants to approximate H 1/2 (Γ) and Dirac deltas for H 1/2 (Γ). The shape of the test functions is controlled by the shifting parameter ǫ and is optimized for third order convergence. In collaboration with Francisco Javier Sayas, Matthew Hassell, Sijiang Lu and Tianyu Qiu, we implemented this idea for the operators associated with Laplace, Helmholtz and Navier-Lamé equations. The code is written in Matlab and is freely available in http://www.math.udel.edu/~fjsayas/deltabem/. Generalizing the ideas behind deltabem to 3D scattering is the subject of ongoing research. Trial Test i 2 i 1 i i+1 i+2 i 2 i 1 i i+1 i+2 i 2 i 1 i i+1 i+2 i 2 i 1 i i+1 i+2 Figure 1: Left: The kite-shaped figure is a cartoon of the staggered grids on the boundary of a scatterer. Open dots represent collocation points and colored crosses represent the symmetrically shifted quadrature points. Center: the shapes for the optimal trial and test functions used as approximations of the trace space (bottom row) and its dual (top row). Right: A possible generalization for 3D scatterers; the equation is collocated at the circles while the red crosses are the location of the optimized quadrature points. Elastic wave propagation and deltabem The analysis of deltabem was recently developed for the operators related to the Helmholtz equation resulting in a one-parameter family of test functions[3]. Nevertheless, for the integral operators associated to the Navier-Lamé resolvent equation (µ( u+ u )+λ ui ) = s 2 u, the extension we developed in [4] was not straightforward due to the two difficulties listed below. 2
First of all, in the elastic case, the double layer operator and its adjoint contain a strongly singular operator (a perturbation of the periodic Hilbert transform) that makes the operators of the second kind much more difficult to handle. In collaboration with Francisco Javier Sayas and Víctor Domínguez [4], we have provided mathematical and numerical evidence that a particular symmetric average of the shiftings ǫ = ±1/6,±5/6 yields third order convergence and is in fact the only choice with this property. The second difficulty was the lack of an appropriate regularization of the hypersingular operator compatible with our method for discretization. We implemented a regularization due to Frangi and Novati [7] which results in an integro-differential operator that is consistent with the deltabem discretization. This provided a complete discretization of the operators of the Calderón Projector that can be used to solve well-posed frequency domain problems or can be combined with Convolution Quadrature for timedomain problems. Figure 2 shows an example of a simulation obtained using this method. Future lines of research in this direction include the analysis of a deltabem discretization of the Calderón projector associated to Stokes s resolvent equations to discretize integral formulations like the one outlined in [1]. Figure 2: A plane pressure wave interacts with a rigid body, the total wave satisfies homogeneous Dirichlet boundary conditions. After the wave is reflected it decomposes into pressure and shear components according to the angle of incidence with respect to the normal. The simulation uses deltabem for space discretization and Trapezoidal Rule Convolution Quadrature for time evolution. Boundary Integral Equation and coupled schemes for wave-structure interaction The study of the scattering of acoustic waves by obstacles with various physical properties and the corresponding reaction of the scatterer is of great importance in the applied sciences. Applications include structural analysis, noise reduction, seismic imaging, and many others. The analysis and simulation of this processes in the time domain can greatly enhance the understanding of the underlying physics by providing additional information related to the transient stages of the problem. In [10, 14, 9] we have addressed the time-domain scattering of acoustic waves by bounded linearly elastic solids, piezoelectric solids and thermoelastic solids. These problems require the coupling of PDE s of hyperbolic, eliptic and parabolic type with non-standard transmission conditions. Table 1 shows the concrete system of PDE s modeling each interaction. In the acoustic/elastic case we have developed two well posed formulations; one based solely on Boundary Integral Equations for homogeneous scatterers, and one coupled Boundary Integral Equation/Variational formulation for general non-homogeneous anisotropic scatterers. For the piezoelectric and thermoelastic cases we have developed symmetric coupling schemes that allow for boundary element treatment of the acoustic wave and a finite element solution for the elastic, electric and thermal unknowns in the solid. The analysis is done in the Laplace domain assuming Lipschitz scatterers and sufficiently smooth incident waves. Well-posedness, at the continuous and semi-discrete-in-space levels, is established by equivalent transmission and variational problems from which stability bounds in terms of the Laplace parameter are derived. Once the systems are discretized in time with Convolution Quadrature, the stability bounds can be translated into time-domain estimates with explicit time dependence. In the short term my goal in this area is to develop integro-differential and boundary-integral formulations of this type for the fluid-structure interaction of elastic bodies of different sorts immersed in 3
a Stokes s flow. Such well posed formulations would allow for the simulation of a wide variety of fluidsolid interactions and, in combination with the computational and analytical tools like deltabem and CQ, would provide a computational toolbox for inexpensive time-domain simulation of a wide array of multiphysics problems. Acoustic/Elastic Acoustic/Piezoelectric Acoustic/Thermoelastic v =c 2 v tt in Ω + v =c 2 v tt in Ω + v =c 2 σ =ρ Σ u tt in Ω σ ζ θ =ρ Σ u tt in Ω v tt in Ω + D =0 in Ω κ θ =(θ +κη u) t in Ω σ =ρ Σ u tt in Ω u t ν = ν(v +v inc u t ν = ν(v +v inc ) on Γ (σ ζθi)ν = ρ f (v +v inc ) tν on Γ ) on Γ σν = ρ f (v +v inc σν = ρ f (v +v inc ) tν on Γ u t ν = ν(v +v inc ) on Γ ) tν on Γ D ν =η d on Γ N νθ =η d on Γ N ψ =µ d on Γ D θ =µ d on Γ D Table 1: Wave-structure Interaction: The equations are posed for t 0 and all the unknowns satisfy homogeneous initial conditions. Here Ω + denotes the unbounded exterior of the scatterer, Ω the interior of the scatterer and Γ the interface which is split into Dirichlet and Neumann parts Γ = Γ D Γ N in the piezoelectric and thermoelastic problems. The density of the elastic body is represented by ρ Σ and the speed of sound in the acoustic domain is c, also v is the unknown scattered acoustic wave, u is the elastic displacement wave and ψ is the electric potential. The stress tensor σ is a function of the displacement u. In piezoelectric solids both σ and the electric displacement vector D depend on a combination of u and ψ, thus coupling the system. The temperature variation with respect to equilibrium is denoted by θ, the thermal diffusivity κ and the thermal expansion coefficient ζ are considered constant. Finite volume methods for non-linear conservation laws In the past I have worked on the numerical treatment of first order non-linear conservation laws. The non-linearity of the problem implies that singularities and discontinuities can develop in the solutions for finite times even for smooth initial data. The classic Godunov-type methods solve this problem by discretizing the initial conditions (for instance averaging on a control volume or cell) and approximating them with piecewise constants or polynomials. The equations are then solved exactly for this approximate initial conditions which requires solving or approximating so-called Riemann problems. Methods of this family are particularly well suited to capture developing shocks in the solution, avoid Gibbs oscillations, and preserve the conservative nature of the continuous problem. For my masters thesis [13] I implemented a FORTRAN code for the solution of Euler s equations of gas dynamics in vacuum t ρ ρu ρv ρw E + x ρu ρu 2 +p ρvu ρwu u(e +p) + y ρv ρuv ρv 2 +p ρwv v(e +p) + z ρw ρuw ρvw ρw 2 +p w(e +p) = 0. Here u,v,w are the components of the velocity, ρ is the gas density, E is the energy and p is the pressure. The 3D implementation was based on dimensional splitting and required the handling of the interfaces with vacuum. It also explored different choices of flux limiters to increase the order of the approximation in the regions far from the shock profiles. Figure 3 shows snapshots of a simulation of a gas sphere colliding with a wall of gas. Recent work has been done [5, 6] on the coupling of Boundary Elements with Finite Volumes for time-independent problems. In the near future I plan to undertake the analysis and implementation of the coupling of these two methods in the time domain. Such couplings would potentially be applied to the time-domain simulation of phenomena such as crack propagation, where the behavior of the fluid in the cracks is traditionally computed on the time-independent regime using finite volumes and the elastic effects are handled with Boundary Elements. 4
Figure 3: Finite volume simulation of a sphere of gas colliding with a wall of gas. The plot shows density iso-surfaces at four different times. The method used dimensional splitting and first order explicit time-stepping. It involved solving a nonlinear Riemann problem at the inteface of each cell for every time step and verifying a positivity condition to handle the generation of vacuum. References [1] C. Bacuta, M. E. Hassell, G. C. Hsiao, and F.-J. Sayas. Boundary integral solvers for an evolutionary exterior stokes problem. SIAM Journal on Numerical Analysis, 53(3):1370 1392, 2015. [2] T. Brown, T. Sánchez-Vizuet, and F.-J. Sayas. Evolution of a semidiscrete system modeling the scattering of acoustic waves by a piezoelectric solid. Submitted, 2016. [3] V. Domínguez, S. Lu, and F.-J. Sayas. A Nyström flavored Calderón calculus of order three for two dimensional waves, time-harmonic and transient. Comput. Math. Appl., 67(1):217 236, 2014. [4] V. Domínguez, T. Sánchez-Vizuet, and F.-J. Sayas. A fully discrete Calderón calculus for the two-dimensional elastic wave equation. Comput. Math. Appl., 69(7):620 635, 2015. [5] C. Erath. Comparison of two couplings of the finite volume method and the boundary element method. In J. Fuhrmann, M. Ohlberger, and C. Rohde, editors, Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects, volume 77 of Springer Proceedings in Mathematics & Statistics, pages 255 263. Springer International Publishing, 2014. [6] C. Erath, G. Of, and F.-J. Sayas. A non-symmetric coupling of the finite volume method and the boundary element method. submitted, 2015. [7] A. Frangi. and G. Novati. Regularized symmetric Galerkin BIE formulations in the Laplace transform domain for 2D problems. Computational Mechanics, 22(1):50 60, 1998. [8] M. Hassell, T. Qiu, T. Sánchez-Vizuet, and F.-J. Sayas. A new and improved analysis of the time domain boundary integral operators for acoustics. To appear in the Journal of Integral Equations and Applications, 2015. [9] G. Hsiao, T. Sánchez-Vizuet, F.-J. Sayas, and R. Weinacht. A time-dependent wave-thermoelastic solid interaction. Submitted, 2016. [10] G. C. Hsiao, T. Sánchez-Vizuet, and F.-J. Sayas. Boundary and coupled boundary-finite element methods for transient wave-structure interaction. IMA J. Numer. Anal., 37(1):237 265, 2016. [11] C. Lubich. Convolution quadrature and discretized operational calculus. I. Numer. Math., 52(2):129 145, 1988. [12] C. Lubich. Convolution quadrature and discretized operational calculus. II. Numer. Math., 52(4):413 425, 1988. [13] T. Sánchez-Vizuet. Numerical solution of Euler s equations of gas dynamics in 3D. Master s thesis, Universidad Nacional Autónoma de México. Instituto de Matemáticas, June 2011. (In Spanish). [14] T. Sánchez-Vizuet and F.-J. Sayas. Symmetric boundary-finite element discretization of time dependent acoustic scattering by elastic obstacles with piezoelectric behavior. Journal of Scientific Computing, 70(3):1290 1315, 2017. [15] J. Saranen and G. Vainikko. Periodic Integral and Pseudodifferential Equations with Numerical Approximation. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2002. 5