An Energy Absorbing Far-Field Boundary Condition for the Elastic Wave Equation

We present an energy absorbing non-reflecting boundary condition of Clayton-Engquist type for the elastic wave equation together with a discretization which is stable for any ratio of compressional to shear wave speed. We prove stability for a second-order accurate finite-difference discretization of the elastic wave equation in three space dimensions together with a discretization of the proposed non-reflecting boundary condition. The stability proof is based on a discrete energy estimate and is valid for heterogeneous materials. The proof includes all six boundaries of the computational domain where special discretizations are needed at the edges and corners. The stability proof holds also when a free surface boundary condition is imposed on some sides of the computational domain.     

Numerical Modeling of Elastic Wave Propagation in a Fluid-Filled Borehole

We review the methods of simulating elastic wave propagation in a borehole. We considered two different approaches: a quasi-analytic approach using the Discrete Wavenumber Summation Method, and the purely numerical Finite Difference Method. We consider the special geometry of the borehole and discuss the problem in cylindrical coordinates. We point out some numerical difficulties that are particularly unique to this problem in cylindrical coordinates.     

A Stable Finite Difference Method for the Elastic Wave Equation on Complex Geometries with Free Surfaces

A stable and explicit second order accurate finite difference method for the elastic wave equation in curvilinear coordinates is presented. The discretization of the spatial operators in the method is shown to be self-adjoint for free-surface, Dirichlet and periodic boundary conditions. The fully discrete version of the method conserves a discrete energy to machine precision.     

Numerical Method for the Deterministic Kardar-Parisi-Zhang Equation in Unbounded Domains

We propose an artificial boundary method for solving the deterministic Kardar-Parisi-Zhang equation in one-, two- and three- dimensional unbounded domains. The exact artificial boundary conditions are obtained on the artificial boundaries. Then the original problems are reduced to equivalent problems in bounded domains. A finite difference method is applied to solve the reduced problems, and some numerical examples are provided to show the effectiveness of the method.     

Numerical Study of Quantized Vortex Interaction in the Ginzburg-Landau Equation on Bounded Domains

In this paper, we study numerically quantized vortex dynamics and their interaction in the two-dimensional (2D) Ginzburg-Landau equation (GLE) with a dimensionless parameter ε>0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition. We begin with a review of the reduced dynamical laws for time evolution of quantized vortex centers in GLE and show how to solve these nonlinear ordinary differential equations numerically. Then we present efficient and accurate numerical methods for discretizing the GLE on either a rectangular or a disk domain under either Dirichlet or homogeneous Neumann boundary condition. Based on these efficient and accurate numerical methods for GLE and the reduced dynamical laws, we simulate quantized vortex interaction of GLE with different ε and under different initial setups including single vortex, vortex pair, vortex dipole and vortex lattice, compare them with those obtained from the corresponding reduced dynamical laws, and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from GLE on vortex interaction. Finally, we also obtain numerically different patterns of the steady states for quantized vortex lattices under the GLE dynamics on bounded domains.     

Large-Scale Electromagnetic Modeling for Multiple Inhomogeneous Domains

We develop a new formulation of the integral equation (IE) method for three-dimensional (3D) electromagnetic (EM) field computation in large-scale models with multiple inhomogeneous domains. This problem arises in many practical applications including modeling the EM fields within the complex geoelectrical structures in geophysical exploration. In geophysical applications, it is difficult to describe an earth structure using the horizontally layered background conductivity model, which is required for the efficient implementation of the conventional IE approach. As a result, a large domain of interest with anomalous conductivity distribution needs to be discretized, which complicates the computations. The new method allows us to consider multiple inhomogeneous domains, where the conductivity distribution is different from that of the background, and to use independent discretizations for different domains. This reduces dramatically the computational resources required for large-scale modeling. In addition, using this method, we can analyze the response of each domain separately without an inappropriate use of the superposition principle for the EM field calculations. The method was carefully tested for the modeling the marine controlled-source electromagnetic (MCSEM) fields for complex geoelectric structures with multiple inhomogeneous domains, such as a seafloor with the rough bathymetry, salt domes, and reservoirs. We have also used this technique to investigate the return induction effects from regional geoelectrical structures, e.g., seafloor bathymetry and salt domes, which can distort the EM response from the geophysical exploration target.     

High-Order Schemes Combining the Modified Equation Approach and Discontinuous Galerkin Approximations for the Wave Equation

We present a new high order method in space and time for solving the wave equation, based on a new interpretation of the "Modified Equation" technique. Indeed, contrary to most of the works, we consider the time discretization before the space discretization. After the time discretization, an additional biharmonic operator appears, which can not be discretized by classical finite elements. We propose a new Discontinuous Galerkin method for the discretization of this operator, and we provide numerical experiments proving that the new method is more accurate than the classical Modified Equation technique with a lower computational burden.     

Numerical Solution of Blow-Up Problems for Nonlinear Wave Equations on Unbounded Domains

The numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered. Applying the unified approach, which is based on the operator splitting method, we construct the efficient nonlinear local absorbing boundary conditions for the nonlinear wave equation, and reduce the nonlinear problem on the unbounded spatial domain to an initial-boundary-value problem on a bounded domain. Then the finite difference method is used to solve the reduced problem on the bounded computational domain. Finally, a broad range of numerical examples are given to demonstrate the effectiveness and accuracy of our method, and some interesting propagation and behaviors of the blow-up problems for nonlinear wave equations are observed.     

Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation

We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation, the first provably convergent discretization and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this two-term regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on usingcontinuous and discrete a priori L^∞ estimates. To provide a high-quality geometric model as input to the AFEM algorithm, we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEM algorithm is also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.     

Multi-Scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains

In this paper, we propose multi-scale deep neural networks (MscaleDNNs)using the idea of radial scaling in frequency domain and activation functions withcompact support. The radial scaling converts the problem of approximation of highfrequency contents of PDEs' solutions to a problem of learning about lower frequencyfunctions, and the compact support activation functions facilitate the separation of frequency contents of the target function to be approximated by corresponding DNNs.As a result, the MscaleDNNs achieve fast uniform convergence over multiple scales.The proposed MscaleDNNs are shown to be superior to traditional fully connectedDNNs and be an effective mesh-less numerical method for Poisson-Boltzmann equations with ample frequency contents over complex and singular domains.     

A Strong Stability-Preserving Predictor-Corrector Method for the Simulation of Elastic Wave Propagation in Anisotropic Media

In this paper, we propose a strong stability-preserving predictor-corrector (SSPC) method based on an implicit Runge-Kutta method to solve the acoustic- and elastic-wave equations. We first transform the wave equations into a system of ordinary differential equations (ODEs) and apply the local extrapolation method to discretize the spatial high-order derivatives, resulting in a system of semi-discrete ODEs. Then we use the SSPC method based on an implicit Runge-Kutta method to solve the semi-discrete ODEs and introduce a weighting parameter into the SSPC method. On top of such a structure, we develop a robust numerical algorithm to effectively suppress the numerical dispersion, which is usually caused by the discretization of wave equations when coarse grids are used or geological models have large velocity contrasts between adjacent layers. Meanwhile, we investigate the performance of the SSPC method including numerical errors and convergence rate, numerical dispersion, and stability criteria with different choices of the weighting parameter to solve 1-D and 2-D acoustic- and elastic-wave equations. When the SSPC is applied to seismic simulations, the computational efficiency is also investigated by comparing the SSPC, the fourth-order Lax-Wendroff correction (LWC) method, and the staggered-grid (SG) finite difference method. Comparisons of synthetic waveforms computed by the SSPC and analytic solutions for acoustic and elastic models are given to illustrate the accuracy and the validity of the SSPC method. Furthermore, several numerical experiments are conducted for the geological models including a 2-D homogeneous transversely isotropic (TI) medium, a two-layer elastic model, and the 2-D SEG/EAGE salt model. The results show that the SSPC can be used as a practical tool for large-scale seismic simulation because of its effectiveness in suppressing numerical dispersion even in the situations such as coarse grids, strong interfaces, or high frequencies.     

A Level Set Method for the Inverse Problem of Wave Equation in the Fluid-Saturated Porous Media

In this paper, a level set method is applied to the inverse problem of 2-D wave equation in the fluid-saturated media. We only consider the situation that the parameter to be recovered takes two different values, which leads to a shape reconstruction problem. A level set function is used to present the discontinuous parameter, and a regularization functional is applied to the level set function for the ill-posed problem. Then the resulting inverse problem with respect to the level set function is solved by using the damped Gauss-Newton method. Numerical experiments show that the method can recover parameter with complicated geometry and the noise in the observation data.     

Clinical Validation of a Predictive Modeling Equation for Sodium

Changes in plasma sodium (Na) concentration during hemodialysis were predicted by changes in Na concentration of the dialysate at equilibrium with the plasma, according to the formula C't = CD - (CD - C'0) [(V0 - QFt)/V0]A/QF, where C'0 and C't are the Na concentration of the dialysate at equilibrium with the plasma at times 0 and t, respectively; QF is the ultrafiltration flow rate; V0 is the initial total body water; and CD is the Na dialysate concentration. This modeling involves only one parameter, A, which is the effective sodium dialysance and depends on the dialyzer, the QF, the plasma water flow rate, and the actual Donnan coefficient. Parameter A was evaluated after 1 h of dialysis. Seven routine 4-h dialysis sessions were performed in which the Na concentration of dialysate at equilibrium with the plasma was measured at varying times. The mean (+/- SEM) difference between predicted and measured values was delta C = 0.5 +/- 0.2 mmol/L. These data support the validity of the model that allows the monitoring of Na dialysate concentration to obtain a prescribed Na plasma concentration at the end of a dialysis session.     

Dimensional Analysis of the Impact of Event Scale Using Structural Equation Modeling

The dimensionality of the Impact of Event Scale (IES) was analyzed using structural equation modeling (SEM). Responses from 321 individuals (62% response) who had experienced a mass murder of seven people 8 months earlier were obtained. A model with a general factor and three subordinate specific factors—Intrusiveness, Avoidance, and Sleep Disturbance—was developed. Scores on the original IES subscales and the SEM factors were related to scores on the General Health Questionnaire. It was concluded that the original subscales could, to a certain degree, be regarded as a reflection of negative affectivity. A more differentiated pattern emerged using the specific latent variables from the SEM analysis.     

Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

We develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902–1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.     

Boundary Integral Modelling of Elastic Wave Propagation in Multi-Layered 2D Media with Irregular Interfaces

We present a semi-analytic method based on the propagation matrix formulation of indirect boundary element method to compute response of elastic (and acoustic) waves in multi-layered media with irregular interfaces. The method works recursively starting from the top-most free surface at which a stress-free boundary condition is applied, and the displacement-stress boundary conditions are then subsequently applied at each interface. The basic idea behind this method is the matrix formulation of the propagation matrix (PM) or more recently the reflectivity method as wide used in the geophysics community for the computation of synthetic seismograms in stratified media. The reflected and transmitted wave-fields between arbitrary shapes of layers can be computed using the indirect boundary element method (BEM, sometimes called IBEM). Like any standard BEM, the primary task of the BEM-based propagation matrix method (thereafter called PM-BEM) is the evaluation of element boundary integral of the Green's function, for which there are standard method that can be adapted. In addition, effective absorbing boundary conditions as used in the finite difference numerical method is adapted in our implementation to suppress the spurious arrivals from the artificial boundaries due to limited model space. To our knowledge, such implementation has not appeared in the literature. We present several examples in this paper to demonstrate the effectiveness of this proposed PM-BEM for modelling elastic waves in media with complex structure.     

The Hamiltonian Field Theory of the Von Mises Wave Equation: Analytical and Computational Issues

The Von Mises quasi-linear second order wave equation, which completely describes an irrotational, compressible and barotropic classical perfect fluid, can be derived from a nontrivial least action principle for the velocity scalar potential only, in contrast to existing analog formulations which are expressed in terms of coupled density and velocity fields. In this article, the classical Hamiltonian field theory specifically associated to such an equation is developed in the polytropic case and numerically verified in a simplified situation. The existence of such a mathematical structure suggests new theoretical schemes possibly useful for performing numerical integrations of fluid dynamical equations. Moreover, it justifies possible new functional forms for Lagrangian densities and associated Hamiltonian functions in other theoretical classical physics contexts.     

An Edge-Based Smoothed Finite Element Method with TBC for the Elastic Wave Scattering by an Obstacle

Elastic wave scattering has received ever-increasing attention in military and medical fields due to its high-precision solution. In this paper, an edge-based smoothed finite element method (ES-FEM) combined with the transparent boundary condition (TBC) is proposed to solve the elastic wave scattering problem by a rigid obstacle with smooth surface, which is embedded in an isotropic and homogeneous elastic medium in two dimensions. The elastic wave scattering problem satisfies Helmholtz equations with coupled boundary conditions obtained by Helmholtz decomposition. Firstly, the TBC of the elastic wave scattering is constructed by using the analytical solution to Helmholtz equations, which can truncate the boundary value problem (BVP) in an unbounded domain into the BVP in a bounded domain. Then the formulations of ES-FEM with the TBC are derived for Helmholtz equations with coupled boundary conditions. Finally, several numerical examples illustrate that the proposed ES-FEM with the TBC (ES-FEM-TBC) can work effectively and obtain more stable and accurate solution than the standard FEM with the TBC (FEM-TBC) for the elastic wave scattering problem.     

A Well-Posed and Discretely Stable Perfectly Matched Layer for Elastic Wave Equations in Second Order Formulation

We present a well-posed and discretely stable perfectly matched layer for the anisotropic (and isotropic) elastic wave equations without first re-writing the governing equations as a first order system. The new model is derived by the complex coordinate stretching technique. Using standard perturbation methods we show that complex frequency shift together with a chosen real scaling factor ensures the decay of eigen-modes for all relevant frequencies. To buttress the stability properties and the robustness of the proposed model, numerical experiments are presented for anisotropic elastic wave equations. The model is approximated with a stable node-centered finite difference scheme that is second order accurate both in time and space.     

Modeling Salt Dependence of Protein-Protein Association: Linear vs Non-Linear Poisson-Boltzmann Equation

Proteins perform various biological functions in the cell by interacting and binding to other proteins, DNA, or other small molecules. These interactions occur in cellular compartments with different salt concentrations, which may also vary under different physiological conditions. The goal of this study is to investigate the effect of salt concentration on the electrostatic component of the binding free energy (hereafter, salt effect) based on a large set of 1482 protein-protein complexes, a task that has never been done before. Since the proteins are irregularly shaped objects, the calculations have been carried out by a means of finite-difference algorithm that solves Poisson-Boltzmann equation (PB) numerically. We performed simulations using both linear and non-linear PB equations and found that non-linearity, in general, does not have significant contribution into salt effect when the net charges of the protein monomers are of different polarity and are less than five electron units. However, for complexes made of monomers carrying large net charges non-linearity is an important factor, especially for homo-complexes which are made of identical units carrying the same net charge. A parameter reflecting the net charge of the monomers is proposed and used as a flag distinguishing between cases which should be treated with non-linear Poisson-Boltzmann equation and cases where linear PB produces sound results. It was also shown that the magnitude of the salt effect is not correlated with macroscopic parameters (such as net charge of the monomers, corresponding complexes, surface and number of interfacial residues) but rather is a complex phenomenon that depends on the shape and charge distribution of the molecules.