Knowledge of helium diffusion kinetics is critical for materials in which helium measurements are made, particulary for thermochronology. In most cases the helium ages were younger than expected, an observation attributes to diffusive loss of helium and the ejection of high energy alpha particles. Therefore it is important to accurately calculate the distribution of the source term within a sample. In this paper, the prediction of the helium concentrations as function of a spatially variable source term are considered. Both the forward and inverse solutions are presented. Under the assumption of radially symmetric geometry, an analytical solution is deduced based on the eigenfunction expansion. Two regularization methods, the Tikhonov regularization and the spectral cutoff regularization, are considered to obtain the regularized solution. Error estimates with optimal convergence order are shown between the exact solution and the regularized solution. Numerical examples are presented to illustrate the validity and effectiveness of the proposed methods

We develop an efficient, adaptive locally weighted projection regression (ALWPR) framework for uncertainty quantification (UQ) of systems governed by ordinary and partial differential equations. The algorithm adaptively selects the new input points with the largest predictive variance and decides when and where to add new local models. It effectively learns the local features and accurately quantifies the uncertainty in the prediction of the statistics. The developed methodology provides predictions and confidence intervals at any query input and can deal with multi-output cases. Numerical examples are presented to show the accuracy and efficiency of the ALWPR framework including problems with non-smooth local features such as discontinuities in the stochastic space.

We present an efficient algorithm for calculating the minimum energy path (MEP) and energy barriers between local minima on a multidimensional potential energy surface (PES). Such paths play a central role in the understanding of transition pathways between metastable states. Our method relies on the original formulation of the string method [Phys. Rev. B, 66,052301 (2002)], i.e. to evolve a smooth curve along a direction normal to the curve. The algorithm works by performing minimization steps on hyperplanes normal to the curve. Therefore the problem of finding MEP on the PES is remodeled as a set of constrained minimization problems. This provides the flexibility of using minimization algorithms faster than the steepest descent method used in the simplified string method [J. Chem. Phys., 126(16), 164103 (2007)]. At the same time, it provides a more direct analog of the finite temperature string method. The applicability of the algorithm is demonstrated using various examples.

Numerical aspects of a pore scale model are investigated for the simulation of catalyst layers of polymer electrolyte membrane fuel cells. Coupled heat, mass and charged species transport together with reaction kinetics are taken into account using parallelized finite volume simulations for a range of nanostructured, computationally reconstructed catalyst layer samples. The effectiveness of implementing deflation as a second stage preconditioner generally improves convergence and results in better convergence behavior than more sophisticated first stage pre-conditioners. This behavior is attributed to the fact that the two stage preconditioner updates the preconditioning matrix at every GMRES restart, reducing the stalling effects that are commonly observed in restarted GMRES when a single stage preconditioner is used. In addition, the effectiveness of the deflation preconditioner is independent of the number of processors, whereas the localized block ILU preconditioner deteriorates in quality as the number of processors is increased. The total number of GMRES search directions required for convergence varies considerably depending on the preconditioner, but also depends on the catalyst layer microstructure, with low porosity microstructures requiring a smaller number of iterations. The improved model and numerical solution strategy should allow simulations for larger computational domains and improve the reliability of the predicted transport parameters. The preconditioning strategies presented in the paper are scalable and should prove effective for massively parallel simulations of other problems involving nonlinear equations.

Due to the rapid advances in micro-electro-mechanical systems (MEMS), the study of microflows becomes increasingly important. Currently, the molecular-based simulation techniques are the most reliable methods for rarefied flow computation, even though these methods face statistical scattering problem in the low speed limit. With discretized particle velocity space, a unified gas-kinetic scheme (UGKS) for entire Knudsen number flow has been constructed recently for flow computation. Contrary to the particle-based direct simulation Monte Carlo (DSMC) method, the unified scheme is a partial differential equation-based modeling method, where the statistical noise is totally removed. But, the common point between the DSMC and UGKS is that both methods are constructed through direct modeling in the discretized space. Due to the multiscale modeling in the unified method, i.e., the update of both macroscopic flow variables and microscopic gas distribution function, the conventional constraint of time step being less than the particle collision time in many direct Boltzmann solvers is released here. The numerical tests show that the unified scheme is more efficient than the particle-based methods in the low speed rarefied flow computation. The main purpose of the current study is to validate the accuracy of the unified scheme in the capturing of non-equilibrium flow phenomena. In the continuum and free molecular limits, the gas distribution function used in the unified scheme for the flux evaluation at a cell interface goes to the corresponding Navier-Stokes and free molecular solutions. In the transition regime, the DSMC solution will be used for the validation of UGKS results. This study shows that the unified scheme is indeed a reliable and accurate flow solver for low speed non-equilibrium flows. It not only recovers the DSMC results whenever available, but also provides high resolution results in cases where the DSMC can hardly afford the computational cost. In thermal creep flow simulation, surprising solution, such as the gas flowing from hot to cold regions along the wall surface, is observed for the first time by the unified scheme, which is confirmed later through intensive DSMC computation.

An existing phase-fieldmodel of two immiscible fluids with a single soluble surfactant present is discussed in detail. We analyze the well-posedness of the model and provide strong evidence that it is mathematically ill-posed for a large set of physically relevant parameters. As a consequence, critical modifications to the model are suggested that substantially increase the domain of validity. Carefully designed numerical simulations offer informative demonstrations as to the sharpness of our theoretical results and the qualities of the physical model. A fully coupled hydrodynamic test-case demonstrates the potential to capture also non-trivial effects on the overall flow.

A novel adaptive approach to compute the eigenenergies and eigenfunctions of the two-particle (electron-hole) Schrödinger equation including Coulomb attraction is presented. As an example, we analyze the energetically lowest exciton state of a thin one-dimensional semiconductor quantum wire in the presence of disorder which arises from the non-smooth interface between the wire and surrounding material. The eigenvalues of the corresponding Schrödinger equation, i.e., the one-dimensional exciton Wannier equation with disorder, correspond to the energies of excitons in the quantum wire. The wavefunctions, in turn, provide information on the optical properties of the wire.

We reformulate the problem of two interacting particles that both can move in one dimension as a stationary eigenvalue problem with two spacial dimensions in an appropriate weak form whose bilinear form is arranged to be symmetric, continuous, and coercive. The disorder of the wire is modelled by adding a potential in the Hamiltonian which is generated by normally distributed random numbers. The numerical solution of this problem is based on adaptive wavelets. Our scheme allows for a convergence proof of the resulting scheme together with complexity estimates. Numerical examples demonstrate the behavior of the smallest eigenvalue, the ground state energies of the exciton, together with the eigenstates depending on the strength and spatial correlation of disorder.

In the first of a series of papers, we will study a discontinuous Galerkin (DG) framework for many electron quantum systems. The salient feature of this framework is the flexibility of using hybrid physics-based local orbitals and accuracy-guaranteed piecewise polynomial basis in representing the Hamiltonian of the many body system. Such a flexibility is made possible by using the discontinuous Galerkin method to approximate the Hamiltonian matrix elements with proper constructions of numerical DG fluxes at the finite element interfaces. In this paper, we will apply the DG method to the density matrix minimization formulation, a popular approach in the density functional theory of many body Schrödinger equations. The density matrix minimization is to find the minima of the total energy, expressed as a functional of the density matrix ρ(r,r′), approximated by the proposed enriched basis, together with two constraints of idempotency and electric neutrality. The idempotency will be handled with the McWeeny’s purification while the neutrality is enforced by imposing the number of electrons with a penalty method. A conjugate gradient method (a Polak-Ribiere variant) is used to solve the minimization problem. Finally, the linear-scaling algorithm and the advantage of using the local orbital enriched finite element basis in the DG approximations are verified by studying examples of one dimensional lattice model systems.

In this article we present a new class of high order accurate Arbitrary-Eulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes. A WENO reconstruction algorithm is used to achieve high order accuracy in space and a high order one-step time discretization is achieved by using the local space-time Galerkin predictor proposed in. For that purpose, a new element-local weak formulation of the governing PDE is adopted on moving space-time elements. The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes. Moreover, a polynomial mapping defined by the same local space-time basis functions as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element. To maintain algorithmic simplicity, the final ALE one-step finite volume scheme uses moving triangular meshes with straight edges. This is possible in the ALE framework, which allows a local mesh velocity that is different from the local fluid velocity. We present numerical convergence rates for the schemes presented in this paper up to sixth order of accuracy in space and time and show some classical numerical test problems for the two-dimensional Euler equations of compressible gas dynamics.

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.

In this article we present a new family of high order accurate Arbitrary Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff hyperbolic balance laws. High order accuracy in space is obtained with a standard WENO reconstruction algorithm and high order in time is obtained using the local space-time discontinuous Galerkin method recently proposed in [20]. In the Lagrangian framework considered here, the local space-time DG predictor is based on a weak formulation of the governing PDE on a moving space-time element. For the space-time basis and test functions we use Lagrange interpolation polynomials defined by tensor-product Gauss-Legendre quadrature points. The moving space-time elements are mapped to a reference element using an isoparametric approach, i.e. the space-time mapping is defined by the same basis functions as the weak solution of the PDE. We show some computational examples in one space-dimension for non-stiff and for stiff balance laws, in particular for the Euler equations of compressible gas dynamics, for the resistive relativistic MHD equations, and for the relativistic radiation hydrodynamics equations. Numerical convergence results are presented for the stiff case up to sixth order of accuracy in space and time and for the non-stiff case up to eighth order of accuracy in space and time.

The development of high-order schemes has been mostly concentrated on the limiters and high-order reconstruction techniques. In this paper, the effect of the flux functions on the performance of high-order schemes will be studied. Based on the same WENO reconstruction, two schemes with different flux functions, i.e., the fifth-order WENO method and the WENO-Gas-kinetic scheme (WENO-GKS), will be compared. The fifth-order finite difference WENO-SW scheme is a characteristic variable reconstruction based method which uses the Steger-Warming flux splitting for inviscid terms, the sixth-order central difference for viscous terms, and three stages Runge-Kutta time stepping for the time integration. On the other hand, the finite volume WENO-GKS is a conservative variable reconstruction based method with the same WENO reconstruction. But, it evaluates a time dependent gas distribution function along a cell interface, and updates the flow variables inside each control volume by integrating the flux function along the boundary of the control volume in both space and time. In order to validate the robustness and accuracy of the schemes, both methods are tested under a wide range of flow conditions: vortex propagation, Mach 3 step problem, and the cavity flow at Reynolds number 3200. Our study shows that both WENO-SW and WENO-GKS yield quantitatively similar results and agree with each other very well provided a sufficient grid resolution is used. With the reduction of mesh points, the WENO-GKS behaves to have less numerical dissipation and present more accurate solutions than those from the WENO-SW in all test cases. For the Navier-Stokes equations, since the WENO-GKS couples inviscid and viscous terms in a single flux evaluation, and the WENO-SW uses an operator splitting technique, it appears that the WENO-SW is more sensitive to the WENO reconstruction and boundary treatment. In terms of efficiency, the finite volume WENO-GKS is about 4 times slower than the finite differenceWENO-SW in two dimensional simulations. The current study clearly shows that besides high-order reconstruction, an accurate gas evolution model or flux function in a high-order scheme is also important in the capturing of physical solutions. In a physical flow, the transport, stress deformation, heat conduction, and viscous heating are all coupled in a single gas evolution process. Therefore, it is preferred to develop such a scheme with multi-dimensionality, and unified treatment of inviscid and dissipative terms. A high-order scheme does prefer a high-order gas evolution model. Even with the rapid advances of high-order reconstruction techniques, the first-order dynamics of the Riemann solution becomes the bottleneck for the further development of high-order schemes. In order to avoid the weakness of the low order flux function, the development of high-order schemes relies heavily on the weak solution of the original governing equations for the update of additional degree of freedom, such as the non-conservative gradients of flow variables, which cannot be physically valid in discontinuous regions.

A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell’s equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in our scheme, discontinuous Galerkin methods are used to discretize not only the spatial domain but also the temporal domain. The proposed numerical scheme is proved to be unconditionally stable, and a convergent rate is established under the L2-norm when polynomials of degree atmost r and k are used for temporal and spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order in time step is observed numerically for the numerical fluxes w.r.t. temporal variable at the grid points.

A straightforward method is presented for computing three-dimensional Stokes flow, due to forces on a surface, with high accuracy at points near the surface. The flow quantities are written as boundary integrals using the free-space Green’s function. To evaluate the integrals near the boundary, the singular kernels are regularized and a simple quadrature is applied in coordinate charts. High order accuracy is obtained by adding special corrections for the regularization and discretization errors, derived here using local asymptotic analysis. Numerical tests demonstrate the uniform convergence rates of the method.

Electromagnetic modeling provides an interesting context to present a link between physical phenomena and homology and cohomology theories. Over the past twenty-five years, a considerable effort has been invested by the computational electromagnetics community to develop fast and general techniques for defining potentials. When magneto-quasi-static discrete formulations based on magnetic scalar potential are employed in problemswhich involve conductive regionswith holes, cuts are needed to make the boundary value problem well defined. While an intimate connection with homology theory has been quickly recognized, heuristic definitions of cuts are surprisingly still dominant in the literature.

The aim of this paper is first to survey several definitions of cuts together with their shortcomings. Then, cuts are defined as generators of the first cohomology group over integers of a finite CW-complex. This provably general definition has also the virtue of providing an automatic, general and efficient algorithm for the computation of cuts. Some counter-examples show that heuristic definitions of cuts should be abandoned. The use of cohomology theory is not an option but the invaluable tool expressly needed to solve this problem.

Hierarchical bases of arbitrary order for (div)-conforming triangular and tetrahedral elements are constructed with the goal of improving the conditioning of the mass and stiffness matrices. For the basis with the triangular element, it is found numerically that the conditioning is acceptable up to the approximation of order four, and is better than a corresponding basis in the dissertation by Sabine Zaglmayr [High Order Finite Element Methods for Electromagnetic Field Computation, Johannes Kepler Universität, Linz, 2006]. The sparsity of the mass matrices from the newly constructed basis and from the one by Zaglmayr is similar for approximations up to order four. The stiffness matrix with the new basis is much sparser than that with the basis by Zaglmayr for approximations up to order four. For the tetrahedral element, it is identified numerically that the conditioning is acceptable only up to the approximation of order three. Compared with the newly constructed basis for the triangular element, the sparsity of the mass matrices from the basis for the tetrahedral element is relatively sparser.

We use front tracking data structures and functions to model the dynamic evolution of fabric surface. We represent the fabric surface by a triangulated mesh with preset equilibrium side length. The stretching and wrinkling of the surface are modeled by the mass-spring system. The external driving force is added to the fabric motion through the “Impulse method” which computes the velocity of the point mass by superposition of momentum. The mass-spring system is a nonlinear ODE system. Added by the numerical and computational analysis, we show that the spring system has an upper bound of the eigen frequency. We analyzed the system by considering two spring models and we proved in one case that all eigenvalues are imaginary and there exists an upper bound for the eigen-frequency This upper bound plays an important role in determining the numerical stability and accuracy of the ODE system. Based on this analysis, we analyzed the numerical accuracy and stability of the nonlinear spring mass system for fabric surface and its tangential and normal motion. We used the fourth order Runge-Kutta method to solve the ODE system and showed that the time step is linearly dependent on the mesh size for the system.

In an interdisciplinary field on mathematics and physics, we examine a physical problem, fluid flow in porous media, which is represented by a stochastic partial differential equation (SPDE). We first give a priori error estimates for the solutions to an optimization problem constrained by the physical model under lower regularity assumptions than the literature. We then use the concept of Galerkin finite element methods to establish a new numerical algorithm to give approximations for our stochastic optimal physical problem. Finally, we develop original computer programs based on the algorithm and use several numerical examples of various situations to see how well our solver works by comparing its outputs to the priori error estimates.

In this paper, we propose a space-time rescaling scheme for computing the long time evolution of multiple precipitates in an elastically stressed medium. The algorithm is second order accurate in time, spectrally accurate in space and enables one to simulate the evolution of precipitates in a fraction of the time normally used by fixed-frame algorithms. In particular, we extend the algorithm recently developed for single particle by Li et al. (Li, Lowengrub and Leo, J. Comput. Phys., 335 (2007), 554) to the multiple particle case, which involves key differences in the method. Our results show that without elasticity there are successive tip splitting phenomena accompanied by the formation of narrow channels between the precipitates. In presence of applied elastic field, the precipitates form dendrite-like structures with the primary arms aligned in the principal directions of the elastic field. We demonstrate that when the far-field flux decreases with the effective radius of the system, tip-splitting and dendrite formation can be suppressed, as in the one particle case. Depending on the initial position of the precipitates, we further observe that some precipitates grow while others may shrink, even when a positive far field flux is applied.

A numerical method based on a homogeneous single-phase flow model is presented to simulate the interaction between pressure wave and flow cavitation. To account for compressibility effects of liquid water, cavitating flow is assumed to be compressible and governed by time-dependent Euler equations with proper equation of state (EOS). The isentropic one-fluid formulation is employed to model the cavitation inception and evolution, while pure liquid phase is modeled by Tait equation of state. Because of large stiffness of Tait EOS and great variation of sound speed in flow field, some of conventional compressible gasdynamics solvers are unstable and even not applicable when extended to calculation of flow cavitation. To overcome the difficulties, a Godunov-type, cell-centered finite volume method is generalized to numerically integrate the governing equations on triangular mesh. The boundary is treated specially to ensure stability of the approach. The method proves to be stable, robust, accurate, time-efficient and oscillation-free.

Novel numerical experiments are designed to investigate unsteady dynamics of the cavitating flow impacted by pressure wave, which is of great interest in engineering applications but has not been studied systematically so far. Numerical simulation indicates that cavity over cylinder can be induced to collapse if the object is accelerated suddenly and extremely high pressure pulse results almost instantaneously. This, however, may be avoided by changing the traveling speed smoothly. The accompanying huge pressure increase may damage underwater devices. However, cavity formed at relatively high upstream speed may be less distorted or affected by shock wave and can recover fully from the initial deformation. It is observed that the cavitating flow starting from a higher freestream velocity is more stable and more resilient with respect to perturbation than the flow with lower background speed. These findings may shed some light on how to control cavitation development to avoid possible damage to operating devices.