Finite element approximations of nonlinear eigenvalue problems in quantum physics

In this paper, we study finite element approximations of a class of nonlinear eigenvalue problems arising from quantum physics. We derive both a priori and a posteriori finite element error estimates and obtain optimal convergence rates for both linear and quadratic finite element approximations. In particular, we analyze the convergence and complexity of an adaptive finite element method. In our analysis, we utilize certain relationship between the finite element eigenvalue problem and the associated finite element boundary value approximations. We also present several numerical examples in quantum physics that support our theory.     

Locally Conservative Continuous Galerkin FEM for Pressure Equation in Two-Phase Flow Model in Subsurfaces

A typical two-phase model for subsurface flow couples the Darcy equation for pressure and a transport equation for saturation in a nonlinear manner. In this paper, we study a combined method consisting of continuous Galerkin finite element methods (CGFEMs) followed by a post-processing technique for Darcy equation and a nodal centered finite volume method (FVM) with upwind schemes for the saturation transport equation, in which the coupled nonlinear problem is solved in the framework of operator decomposition. The post-processing technique is applied to CGFEM solutions to obtain locally conservative fluxes which ensures accuracy and robustness of the FVM solver for the saturation transport equation. We applied both upwind scheme and upwind scheme with slope limiter for FVM on triangular meshes in order to eliminate the non-physical oscillations. Various numerical examples are presented to demonstrate the performance of the overall methodology.     

A Nitsche-extended finite element method for earthquake rupture on complex fault systems

The extended finite element method (XFEM) provides a natural way to incorporate strong and weak discontinuities into discretizations. It alleviates the need to mesh discontinuities, allowing simulation meshes to be nearly independent of discontinuity geometry. Currently, both quasistatic deformation and dynamic earthquake rupture simulations under standard FEM are limited to simplified fault networks, as generating meshes that both conform with the faults and have appropriate properties for accurate simulation is a difficult problem. In addition, fault geometry is not well known; robustness of solution to fault geometry must be determined. Remeshing with varying geometry would make such tests computationally unfeasible. The XFEM makes a natural choice for discretization in these crustal deformation simulations on complex fault systems. Here, we develop a method based upon the XFEM using Nitsche’s method to apply boundary conditions, enabling the solution of static deformation and dynamic earthquake models. We compare several approaches to calculating and applying frictional tractions. Finally, we demonstrate the method with two problems: an earthquake community dynamic code verification benchmark and a quasistatic problem on a fault system model of southern California.     

A Solution Method for a Certain Nonlinear Interface Problem in Unbounded Domains

In this paper, we consider a kind of nonlinear interface problem in unbounded domains. To solve this problem, we discuss a new coupling of finite element and boundary element by adding an auxiliary circle. We first derive the optimal error estimate of finite element approximation to the coupled FEM-BEM problem. Then we introduce a preconditioning steepest descent method for solving the discrete system by constructing a cheap domain decomposition preconditioner. Moreover, we give a complete analysis to the convergence speed of this iterative method. Received March 30, 2000; revised November 29, 2000     

Computational modeling of electrochemical coupling: A novel finite element approach towards ionic models for cardiac electrophysiology

We propose a novel, efficient finite element solution technique to simulate the electrochemical response of excitable cardiac tissue. We apply a global–local split in which the membrane potential of the electrical problem is introduced globally as a nodal degree of freedom, while the state variables of the chemical problem are treated locally as internal variables on the integration point level. This particular discretization is efficient and highly modular since different cardiac cell models can be incorporated in a straightforward way through only minor local modifications on the constitutive level. Here, we derive the underlying algorithmic framework for a recently proposed ionic model for human ventricular cardiomyocytes, and demonstrate its integration into an existing nonlinear finite element infrastructure. To ensure unconditional algorithmic stability, we apply an implicit backward Euler scheme to discretize the evolution equations for both the electrical potential and the chemical state variables in time. To increase robustness and guarantee optimal quadratic convergence, we suggest an incremental iterative Newton–Raphson scheme and illustrate the consistent linearization of the weak form of the excitation problem. This particular solution strategy allows us to apply an adaptive time stepping scheme, which automatically generates small time steps during the rapid upstroke, and large time steps during the plateau, the repolarization, and the resting phases. We demonstrate that solving an entire cardiac cycle for a real patient-specific geometry characterized through a transmembrane potential, four ion concentrations, thirteen gating variables, and fifteen ionic currents requires computation times of less than ten minutes on a standard desktop computer.     

A combined finite and infinite element approach for modeling spherically symmetric transient subsurface flow

This paper presents a finite element-infinite element coupling approach for modeling a spherically symmetric transient flow problem in a porous medium of infinite extent. A finite element model is used to examine the flow potential distribution in a truncated bounded region close to the spherical cavity. In order to give an appropriate artificial boundary condition at the truncated boundary, a transient infinite element, that is developed to describe transient flow in the exterior unbounded domain, is coupled with the finite element model. The coupling procedure of the finite and infinite elements at their interface is described by means of the boundary integro-differential equation rather than through a matrix approach. Consequently, a Neumann boundary condition can be applied at the truncated boundary to ensure the C¹-continuity of the solution at the truncated boundary. Numerical analyses indicate that the proposed finite element-infinite element coupling approach can generate a correct artificial truncated boundary condition to the finite element model for the unbounded flow transport problem.     

Solid Geometry Processing on Deconstructed Domains

Many tasks in geometry processing are modelled as variational problems solved numerically using the finite element method. For solid shapes, this requires a volumetric discretization, such as a boundary conforming tetrahedral mesh. Unfortunately, tetrahedral meshing remains an open challenge and existing methods either struggle to conform to complex boundary surfaces or require manual intervention to prevent failure. Rather than create a single volumetric mesh for the entire shape, we advocate for solid geometry processing on deconstructed domains, where a large and complex shape is composed of overlapping solid subdomains. As each smaller and simpler part is now easier to tetrahedralize, the question becomes how to account for overlaps during problem modelling and how to couple solutions on each subdomain together algebraically. We explore how and why previous coupling methods fail, and propose a method that couples solid domains only along their boundary surfaces. We demonstrate the superiority of this method through empirical convergence tests and qualitative applications to solid geometry processing on a variety of popular second‐order and fourth‐order partial differential equations.     

Finite Element Approximation to a Finite-Size Modified Poisson-Boltzmann Equation

The inclusion of steric effects is important when determining the electrostatic potential near a solute surface. We consider a modified form of the Poisson-Boltzmann equation, often called the Poisson-Bikerman equation, in order to model these effects. The modifications lead to bounded ionic concentration profiles and are consistent with the Poisson-Boltzmann equation in the limit of zero-size ions. Moreover, the modified equation fits well into existing finite element frameworks for the Poisson-Boltzmann equation. In this paper, we advocate a wider use of the modified equation and establish well-posedness of the weak problem along with convergence of an associated finite element formulation. We also examine several practical considerations such as conditioning of the linearized form of the nonlinear modified Poisson-Boltzmann equation, implications in numerical evaluation of the modified form, and utility of the modified equation in the context of the classical Poisson-Boltzmann equation.     

Tailored Finite Point Method for a Singular Perturbation Problem with Variable Coefficients in Two Dimensions

In this paper, we propose a tailored-finite-point method for a type of linear singular perturbation problem in two dimensions. Our finite point method has been tailored to some particular properties of the problem. Therefore, our new method can achieve very high accuracy with very coarse mesh even for very small ε, i.e. the boundary layers and interior layers do not need to be resolved numerically. In our numerical implementation, we study the classification of all the singular points for the corresponding degenerate first order linear dynamic system. We also study some cases with nonlinear coefficients. Our tailored finite point method is very efficient in both linear and nonlinear coefficients cases.     

Numerical Analysis of Second Order,Fully Discrete Energy Stable Schemes for Phase Field Models of Two-Phase Incompressible Flows

In this paper, we propose several second order in time, fully discrete, linear and nonlinear numerical schemes for solving the phase field model of two-phase incompressible flows, in the framework of finite element method. The schemes are based on the second order Crank–Nicolson method for time discretization, projection method for Navier–Stokes equations, as well as several implicit–explicit treatments for phase field equations. The energy stability and unique solvability of the proposed schemes are proved. Ample numerical experiments are performed to validate the accuracy and efficiency of the proposed schemes.     

Computational model of three-dimensional cardiac electromechanics

We coupled a continuum model of impulse propagation with a three-dimensional model of regional ventricular mechanics. Equations for action potential propagation, myofilament activation and active contraction were solved in an anatomically detailed finite element model of canine left and right ventricular geometry, muscle fiber architecture and Purkinje fiber network anatomy. Finite element equations for time-dependent excitation and recovery variables were assembled using a collocation method and solved using adaptive Runge-Kutta integration. Resting tissue mechanics were modeled as nonlinear, orthotropic and hyperelastic. A Windkessel model for arterial impedance was coupled to ventricular pressure and volume to compute hemodynamic boundary conditions during ejection. Ventricular volume constraints were imposed during the isovolumic phases. This model showed good agreement in the time course of regional systolic strains with experimental measurements during normal sinus rhythm and demonstrates the importance of the Purkinje fiber system in determining the mechanical activation sequence.     

An efficient method for solving elliptic boundary element problems with application to the tokamak vacuum problem

A method for regularizing ill-posed Neumann Poisson-type problems based on applying operator transformations is presented. This method can be implemented in the context of the finite element method to compute the solution to inhomogeneous Neumann boundary conditions; it allows to overcome cases where the Neumann problem otherwise admits an infinite number of solutions. As a test application, we solve the Grad–Shafranov boundary problem in a toroidally symmetric geometry. Solving the regularized Neumann response problem is found to be several orders of magnitudes more efficient than solving the Dirichlet problem, which makes the approach competitive with the boundary element method without the need to derive a Green function. In the context of the boundary element method, the operator transformation technique can also be applied to obtain the response of the Grad–Shafranov equation from the toroidal Laplace n=1 response matrix using a simple matrix transformation.     

Numerical comparison between two cost functions in contact shape optimization

A finite element approach to shape optimization in a 2D frictionless contact problem for two different cost functions is presented in this work. The goal is to find an appropriate shape for the contact boundary, performing an almost constant contact-stress distribution. The whole formulation, including the mathematical model for the unilateral problem, sensitivity analysis and geometry definition is treated in a continuous form, independently of the discretization in finite elements. Shape optimization is performed by a direct modification of the geometry throughB-spline curves and an automatic mesh generator is used at each new configuration to provide the finite element input data. Augmented-Lagrangian techniques (to solve the contact problem) and an interior-point mathematical-programming algorithm (for shape optimization) are used to obtain numerical results.     

Computational models for weakly dispersive nonlinear water waves

Numerical methods for the two- and three-dimensional Boussinesq equations governing weakly nonlinear and dispersive water waves are presented and investigated. Convenient handling of grids adapted to the geometry or bottom topography is enabled by finite element discretization in space. Staggered finite difference schemes are used for the temporal discretization, resulting in only two linear systems to be solved during each time step. Efficient iterative solution of linear systems is discussed. By introducing correction terms in the equations, a fourth-order, two-level temporal scheme can be obtained. Combined with (bi-) quadratic finite elements, the truncation errors of this scheme can be made of the same order as the neglected perturbation terms in the analytical model, provided that the element size is of the same order as the characteristic depth. We present analysis of the proposed schemes in terms of numerical dispersion relations. Verification of the schemes and their implementations is performed for standing waves in a closed basin with constant depth. More challenging applications cover plane incoming waves on a curved beach and earthquake induced waves over a shallow seamount. In the latter example we demonstrate a significantly increased computational efficiency when using higher-order schemes and bathymetry-adapted finite element grids.     

Designing plates for minimum internal resonances

Internal resonance is a nonlinear phenomenon for a structure when the eigenfrequencies of the structure are commensurable or close to being commensurable. Using optimization we have the possibility to control the eigenfrequencies, i.e., move an eigenfrequency, maximize a given eigenfrequency, or maximize the gap between eigenfrequencies. It is therefore also possible to design a structure that is as free as possible of internal resonance up to mode of order n.We consider plates made of two materials. The designs depend on the boundary conditions and on the frequency range within which the plate should be as free of internal resonance as possible. The two materials can either be two physical materials, or one can be a physical material and the other a weakening of the first material. By doing this we are in principle solving three different problems: a reinforcement problem, a problem of where to put holes in the structure, and, finally the more involved case (from a manufacturing point of view), of two different materials. The optimizations are performed using the finite element method for analysis and the topology optimization approach for design. The optimization problem is formulated using a bound formulation where the objective is to maximize a minimum detuning parameter. Special attention is given to the formulation of the conditions for internal resonance. Using the method presented in this paper it is possible to remove an unwanted nonlinear phenomenon without the use of a nonlinear model and without knowledge of the nonlinearities present in the system.     

Parameter estimation approach to the free boundary for the pricing of an american call option

In this paper, we consider a free boundary problem which arises in the pricing of an American call option. The free boundary represents the optimal exercise price as a function of time before a maturity date. We are developing a parameter estimation technique to obtain both the optimal exercise curve of an American call option and its price. For the numerical solution of a forward problem, a time marching finite element method is adopted. Numerical experiment shows the convergence property of the approximation scheme.     

Asymptotic boundary conditions with immersed finite elements for interface magnetostatic/electrostatic field problems with open boundary

Many of the magnetostatic/electrostatic field problems encountered in aerospace engineering, such as plasma sheath simulation and ion neutralization process in space, are not confined to finite domain and non-interface problems, but characterized as open boundary and interface problems. Asymptotic boundary conditions (ABC) and immersed finite elements (IFE) are relatively new tools to handle open boundaries and interface problems respectively. Compared with the traditional truncation approach, asymptotic boundary conditions need a much smaller domain to achieve the same accuracy. When regular finite element methods are applied to an interface problem, it is necessary to use a body-fitting mesh in order to obtain the optimal convergence rate. However, immersed finite elements possess the same optimal convergence rate on a Cartesian mesh, which is critical to many applications. This paper applies immersed finite element methods and asymptotic boundary conditions to solve an interface problem arising from electric field simulation in composite materials with open boundary. Numerical examples are provided to demonstrate the high global accuracy of the IFE method with ABC based on Cartesian meshes, especially around both interface and boundary. This algorithm uses a much smaller domain than the truncation approach in order to achieve the same accuracy.     

A mixed–primal finite element method for the Boussinesq problem with temperature-dependent viscosity

In this paper we focus on the analysis of a mixed finite element method for a class of natural convection problems in two dimensions. More precisely, we consider a system based on the coupling of the steady-state equations of momentum (Navier–Stokes) and thermal energy by means of the Boussinesq approximation (coined the Boussinesq problem), where we also take into account a temperature dependence of the viscosity of the fluid. The construction of this finite element method begins with the introduction of the pseudostress and vorticity tensors, and a mixed formulation for the momentum equations, which is augmented with Galerkin-type terms, in order to deal with the non-linearity of these equations and the convective term in the energy equation, where a primal formulation is considered. The prescribed temperature on the boundary becomes an essential condition, which is weakly imposed, leading us to the definition of the normal heat flux through the boundary as a Lagrange multiplier. We show that this highly coupled problem can be uncoupled and analysed as a fixed-point problem, where Banach and Brouwer theorems will help us to provide sufficient conditions to ensure well-posedness of the problems arising from the continuous and discrete formulations, along with several applications of continuous injections guaranteed by the Rellich–Kondrachov theorem. Finally, we show some numerical results to illustrate the performance of this finite element method, as well as to prove the associated rates of convergence.     

On the adaptive finite element method of steady-state rolling contact for hyperelasticity in finite deformations

In this paper, we present an adaptive finite element method for steady-state rolling contact in finite deformations along with a residual based a posteriori error estimator for rolling contact problem with Coulomb friction. A general formulation of rolling contact geometry is derived from the point of view of differential geometry. Solvability conditions for the rolling contact problems are discussed. We use Newton's method to solve variational equations derived from a penalty regularization of the finite element approximation of the rolling contact problem. We provide a numerical example to illustrate the method.