Exact Artificial Boundary Condition for the Poisson Equation in the Simulation of the 2D Schrödinger-Poisson System

We study the computation of ground states and time dependent solutions of the Schrödinger-Poisson system (SPS) on a bounded domain in 2D (i.e. in two space dimensions). On a disc-shaped domain, we derive exact artificial boundary conditions for the Poisson potential based on truncated Fourier series expansion in θ, and propose a second order finite difference scheme to solve the $r$-variable ODEs of the Fourier coefficients. The Poisson potential can be solved within $\mathcal{O}$($M NlogN$) arithmetic operations where $M,N$ are the number of grid points in $r$-direction and the Fourier bases. Combined with the Poisson solver, a backward Euler and a semi-implicit/leap-frog method are proposed to compute the ground state and dynamics respectively. Numerical results are shown to confirm the accuracy and efficiency. Also we make it clear that backward Euler sine pseudospectral (BESP) method in [33] can not be applied to 2D SPS simulation.     

Numerical Discretization of Variational Phase Field Model for Phase Transitions in Ferroelectric Thin Films

Phase field methods have been widely used to study phase transitions and polarization switching in ferroelectric thin films. In this paper, we develop an efficient numerical scheme for the variational phase field model based on variational forms of the electrostatic energy and the relaxation dynamics of the polarization vector. The spatial discretization combines the Fourier spectral method with the finite difference method to handle three-dimensional mixed boundary conditions. It allows for an efficient semi-implicit discretization for the time integration of the relaxation dynamics. This method avoids explicitly solving the electrostatic equilibrium equation (a Poisson equation) and eliminates the use of associated Lagrange multipliers. We present several numerical examples including phase transitions and polarization switching processes to demonstrate the effectiveness of the proposed method.     

Numerical Boundary Conditions for Specular Reflection in a Level-Sets-Based Wavefront Propagation Method

We study the simulation of specular reflection in a level set method implementation for wavefront propagation in high frequency acoustics using WENO spatial operators. To implement WENO efficiently and maintain convergence rate, a rectangular grid is used over the physical space. When the physical domain does not conform to the rectangular grid, appropriate boundary conditions to represent reflection must be derived to apply at grid locations that are not coincident with the reflecting boundary. A related problem is the extraction of the normal vectors to the boundary, which is required to formulate the reflection condition. A separate level set method is applied to pre-compute the boundary normals which are then stored for use in the wavefront method. Two approaches to handling the reflection boundary condition are proposed and studied: one uses an approximation to the boundary location, and the other uses a local reflection principle. The second method is shown to produce superior results.     

Validation of the coupling of magnetic resonance imaging velocity measurements with computational fluid dynamics in a U bend

Magnetic resonance imaging (MRI) can be used in vivo in combination with computational fluid dynamics (CFD) to derive velocity profiles in space and time and accordingly, pressure drop and wall shear stress distribution in natural or artificial vessel segments. These hemodynamic data are difficult or impossible to acquire directly in vivo. Therefore, research has been performed combining MRI and CFD for flow simulations in flow phantoms, such as bends or anastomoses, and even in human vessels such as the aorta, the carotid, and the abdominal bifurcation. There is, however, no unanimity concerning the use of MRI velocity measurements as input for the inflow boundary condition of a CFD simulation. In this study, different input possibilities for the inflow boundary conditions are compared. MRI measurements of steady and pulsatile flow were performed on a U bend phantom, representing the aorta geometry. PAMFLOW (ESI Software, Krimpen aan den Ussel, The Netherlands), an industrial CFD software package, was used to solve the Navier-Stokes equations for incompressible flow. Three main parameters were found to influence the choice of an inflow boundary condition type. First, the flow rate through a vessel should be exact, since it proves to be a determining factor for the accuracy of the velocity profile. The other decisive parameters are the physiology of the flow profile and the required computer processing unit time. Our comparative study indicates that the best way to handle an inflow boundary condition is to use the velocities measured by MRI at the inflow plane as being fixed velocities. However, before using these MRI velocity data, they first should be corrected for the partial volume effect by filtering and second scaled in order to obtain the correct flow rate. This implies that a reliable flow rate measurement absolutely is needed for CFD calculations based on MRI velocity measurements.     

Study of Simple Hydrodynamic Solutions with the Two-Relaxation-Times Lattice Boltzmann Scheme

For simple hydrodynamic solutions, where the pressure and the velocity are polynomial functions of the coordinates, exact microscopic solutions are constructed for the two-relaxation-time (TRT) Lattice Boltzmann model with variable forcing and supported by exact boundary schemes. We show how simple numerical and analytical solutions can be interrelated for Dirichlet velocity, pressure and mixed (pressure/tangential velocity) multi-reflection (MR) type schemes. Special care is taken to adapt them for corners, to examine the uniqueness of the obtained steady solutions and staggered invariants, to validate their exact parametrization by the non-dimensional hydrodynamic and a "kinetic" (collision) number. We also present an inlet/outlet "constant mass flux" condition. We show, both analytically and numerically, that the kinetic boundary schemes may result in the appearance of Knudsen layers which are beyond the methodology of the Chapman-Enskog analysis. Time dependent Dirichlet boundary conditions are investigated for pulsatile flow driven by an oscillating pressure drop or forcing. Analytical approximations are constructed in order to extend the pulsatile solution for compressible regimes.     

Exact penalty functions for optimal control problems I: Main theorem and free‐endpoint problems

In this two‐part study, we develop a general approach to the design and analysis of exact penalty functions for various optimal control problems, including problems with terminal and state constraints, problems involving differential inclusions, and optimal control problems for linear evolution equations. This approach allows one to simplify an optimal control problem by removing some (or all) constraints of this problem with the use of an exact penalty function, thus allowing one to reduce optimal control problems to equivalent variational problems and apply numerical methods for solving, eg, problems without state constraints, to problems including such constraints, etc. In the first part of our study, we strengthen some existing results on exact penalty functions for optimisation problems in infinite dimensional spaces and utilise them to study exact penalty functions for free‐endpoint optimal control problems, which reduce these problems to equivalent variational ones. We also prove several auxiliary results on integral functionals and Nemytskii operators that are helpful for verifying the assumptions under which the proposed penalty functions are exact.     

Accurate Boundary Conditions for Twin Boundary

In this paper, we propose accurate numerical boundary conditions for atomic simulations of twin boundary. The heterogeneity of the lattice structure induces physical reflection across the twin boundary. When numerical boundary and the twin boundary coincide, the goal is to reproduce the correct amount of physical reflection. In particular, we consider waves periodic in the direction parallel to the twin boundary and reduce the problem into a complex-valued chain motion. Using Laplace transform, we design time history kernel (THK) treatment. We further design matching boundary conditions (MBC) by reproducing physical reflection at long wave limit and a specific wave number. Reflection analysis and numerical tests demonstrate the effectiveness of the proposed THK and MBC treatments.     

Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions

We propose a new method to deal with the essential boundary conditions encountered in the deep learning-based numerical solvers for partial differential equations. The trial functions representing by deep neural networks are non-interpolatory, which makes the enforcement of the essential boundary conditions a nontrivial matter. Our method resorts to Nitsche's variational formulation to deal with this difficulty, which is consistent, and does not require significant extra computational costs. We prove the error estimate in the energy norm and illustrate the method on several representative problems posed in at most 100 dimension.     

Phase Field Model of Thermo-Induced Marangoni Effects in the Mixtures and Its Numerical Simulations with Mixed Finite Element Method

In this paper, we study the Marangoni effects in the mixture of two Newtonian fluids due to the thermo-induced surface tension heterogeneity on the interface. We employ an energetic variational phase field model to describe its physical phenomena, and obtain the corresponding governing equations defined by a modified Navier-Stokes equations coupled with phase field and energy transport. A mixed Taylor-Hood finite element discretization together with full Newton's method are applied to this strongly nonlinear phase field model on a specific domain. Under different boundary conditions of temperature, the resulting numerical solutions illustrate that the thermal energy plays a fundamental role in the interfacial dynamics of two-phase flows. In particular, it gives rise to a dynamic interfacial tension that depends on the direction of temperature gradient, determining the movement of the interface along a sine/cosine-like curve.     

On a control problem governed by a linear partial differential equation with a smooth functional

In this paper, the optimal control problem for the Helmholtz equation with non‐local boundary conditions is considered. The necessary and sufficient conditions of optimality in a maximum principle form have been obtained. We note that this problem is basically different from classical‐type problems because it is impossible to use Green's formula and we cannot rewrite it in the variational form widely used in the literature. So it is impossible to use all the theory that has been developed for optimal control problems with classical boundary conditions. Copyright © 2009 John Wiley & Sons, Ltd.     

A Novel Iterative Penalty Method to Enforce Boundary Conditions in Finite Volume POD-Galerkin Reduced Order Models for Fluid Dynamics Problems

A Finite-Volume based POD-Galerkin reduced order model is developed for fluid dynamics problems where the (time-dependent) boundary conditions are controlled using two different boundary control strategies: the lifting function method, whose aim is to obtain homogeneous basis functions for the reduced basis space and the penalty method where the boundary conditions are enforced in the reduced order model using a penalty factor. The penalty method is improved by using an iterative solver for the determination of the penalty factor rather than tuning the factor with a sensitivity analysis or numerical experimentation.The boundary control methods are compared and tested for two cases: the classical lid driven cavity benchmark problem and a Y-junction flow case with two inlet channels and one outlet channel. The results show that the boundaries of the reduced order model can be controlled with the boundary control methods and the same order of accuracy is achieved for the velocity and pressure fields. Finally, the reduced order models are 270-308 times faster than the full order models for the lid driven cavity test case and 13-24 times for the Y-junction test case.     

Truncation Errors,Exact and Heuristic Stability Analysis of Two-Relaxation-Times Lattice Boltzmann Schemes for Anisotropic Advection-Diffusion Equation

This paper establishes relations between the stability and the high-order truncated corrections for modeling of the mass conservation equation with the two-relaxation-times (TRT) collision operator. First we propose a simple method to derive the truncation errors from the exact, central-difference type, recurrence equations of the TRT scheme. They also supply its equivalent three-time-level discretization form. Two different relationships of the two relaxation rates nullify the third (advection) and fourth (pure diffusion) truncation errors, for any linear equilibrium and any velocity set. However, the two relaxation times alone cannot remove the leading-order advection-diffusion error, because of the intrinsic fourth-order numerical diffusion. The truncation analysis is carefully verified for the evolution of concentration waves with the anisotropic diffusion tensors. The anisotropic equilibrium functions are presented in a simple but general form, suitable for the minimal velocity sets and the d2Q9, d3Q13, d3Q15 and d3Q19 velocity sets. All anisotropic schemes are complemented by their exact necessary von Neumann stability conditions and equivalent finite-difference stencils. The sufficient stability conditions are proposed for the most stable (OTRT) family, which enables modeling at any Peclet numbers with the same velocity amplitude. The heuristic stability analysis of the fourth-order truncated corrections extends the optimal stability to larger relationships of the two relaxation rates, in agreement with the exact (one-dimensional) and numerical (multi-dimensional) stability analysis. A special attention is put on the choice of the equilibrium weights. By combining accuracy and stability predictions, several strategies for selecting the relaxation and free-tunable equilibrium parameters are suggested and applied to the evolution of the Gaussian hill.     

Hamiltonian two‐degrees‐of‐freedom control of chemical reactors

A novel unified approach to two‐degrees‐of‐freedom control is devised and applied to a classical chemical reactor model. The scheme is constructed from the optimal control point of view and along the lines of the Hamiltonian formalism for nonlinear processes. The proposed scheme optimizes both the feedforward and the feedback components of the control variable with respect to the same cost objective. The original Hamiltonian function governs the feedforward dynamics, and its derivatives are part of the gain for the feedback component. The optimal state trajectory is generated online, and is tracked by a combination of deterministic and stochastic optimal tools. The relevant numerical data to manipulate all stages come from a unique off‐line calculation, which provides design information for a whole family of related control problems. This is possible because a new set of PDEs (the variational equations) allow to recover the initial value of the costate variable, and the Hamilton equations can then be solved as an initial‐value problem. Perturbations from the optimal trajectory are abated through an optimal state estimator and a deterministic regulator with a generalized Riccati gain. Both gains are updated online, starting with initial values extracted from the solution to the variational equations. The control strategy is particularly useful in driving nonlinear processes from an equilibrium point to an arbitrary target in a finite‐horizon optimization context. Copyright © 2010 John Wiley & Sons, Ltd.     

An Exact Absorbing Boundary Condition for the Schrödinger Equation with Sinusoidal Potentials at Infinity

In this paper we study numerical issues related to the Schrödinger equation with sinusoidal potentials at infinity. An exact absorbing boundary condition in a form of Dirichlet-to-Neumann mapping is derived. This boundary condition is based on an analytical expression of the logarithmic derivative of the Floquet solution to Mathieu's equation, which is completely new to the author's knowledge. The implementation of this exact boundary condition is discussed, and a fast evaluation method is used to reduce the computation burden arising from the involved half-order derivative operator. Some numerical tests are given to show the performance of the proposed absorbing boundary conditions.     

An Augmented Lagrangian Deep Learning Method for Variational Problems with Essential Boundary Conditions

This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions. To this end, we first reformulate the original problem into a minimax problem corresponding to a feasible augmented Lagrangian, which can be solved by the augmented Lagrangian method in an infinite dimensional setting. Based on this, by expressing the primal and dual variables with two individual deep neural network functions, we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimization method together with a projection technique. Compared to the traditional penalty method, the new method admits two main advantages: i) the choice of the penalty parameter is flexible and robust, and ii) the numerical solution is more accurate in the same magnitude of computational cost. As typical applications, we apply the new approach to solve elliptic problems and (nonlinear) eigenvalue problems with essential boundary conditions, and numerical experiments are presented to show the effectiveness of the new method.     

A Variational Model for Two-Phase Immiscible Electroosmotic Flow at Solid Surfaces

We develop a continuum hydrodynamic model for two-phase immiscible flows that involve electroosmotic effect in an electrolyte and moving contact line at solid surfaces. The model is derived through a variational approach based on the Onsager principle of minimum energy dissipation. This approach was first presented in the derivation of a continuum hydrodynamic model for moving contact line in neutral two-phase immiscible flows (Qian, Wang, and Sheng, J. Fluid Mech. 564, 333–360 (2006)). Physically, the electroosmotic effect can be formulated by the Onsager principle as well in the linear response regime. Therefore, the same variational approach is applied here to the derivation of the continuum hydrodynamic model for charged two-phase immiscible flows where one fluid component is an electrolyte exhibiting electroosmotic effect on a charged surface. A phase field is employed to model the diffuse interface between two immiscible fluid components, one being the electrolyte and the other a nonconductive fluid, both allowed to slip at solid surfaces. Our model consists of the incompressible Navier-Stokes equation for momentum transport, the Nernst-Planck equation for ion transport, the Cahn-Hilliard phase-field equation for interface motion, and the Poisson equation for electric potential, along with all the necessary boundary conditions. In particular, all the dynamic boundary conditions at solid surfaces, including the generalized Navier boundary condition for slip, are derived together with the equations of motion in the bulk region. Numerical examples in two-dimensional space, which involve overlapped electric double layer fields, have been presented to demonstrate the validity and applicability of the model, and a few salient features of the two-phase immiscible electroosmotic flows at solid surface. The wall slip in the vicinity of moving contact line and the Smoluchowski slip in the electric double layer are both investigated.     

Wavelet Galerkin Methods for Aerosol Dynamic Equations in Atmospheric Environment

Aerosol modelling is very important to study and simulate the behavior of aerosol dynamics in atmospheric environment. In this paper, we consider the general nonlinear aerosol dynamic equations which describe the evolution of the aerosol distribution. Continuous time and discrete time wavelet Galerkin methods are proposed for solving this problem. By using the Schauder's fixed point theorem and the variational technique, the global existence and uniqueness of solution of continuous time wavelet numerical methods are established for the nonlinear aerosol dynamics with sufficiently smooth initial conditions. Optimal error estimates are obtained for both continuous and discrete time wavelet Galerkin schemes. Numerical examples are given to show the efficiency of the wavelet technique     

Random Batch Algorithms for Quantum Monte Carlo Simulations

Random batch algorithms are constructed for quantum Monte Carlo simulations. The main objective is to alleviate the computational cost associated with the calculations of two-body interactions, including the pairwise interactions in the potential energy, and the two-body terms in the Jastrow factor. In the framework of variational Monte Carlo methods, the random batch algorithm is constructed based on the over-damped Langevin dynamics, so that updating the position of each particle in an $N$-particle system only requires$\mathcal{O}(1)$ operations, thus for each time step the computational cost for $N$ particles is reduced from$\mathcal{O}(N^2)$ to$\mathcal{O}(N)$. For diffusion Monte Carlo methods, the random batch algorithm uses an energy decomposition to avoid the computation of the total energy in the branching step. The effectiveness of the random batch method is demonstrated using a system of liquid $^4$He atoms interacting with a graphite surface.     

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.     

Hypophosphatemic rickets is associated with disruption of mineral orientation at the nanoscale in the flat scapula bones of rachitic mice with development

Metabolic bone disorders such as rickets are associated with altered in vivo muscular force distributions on the skeletal system. During development, these altered forces can potentially affect the spatial and temporal dynamics of mineralised tissue formation, but the exact mechanisms are not known. Here we have used a murine model of hypophosphatemic rickets (Hpr) to study the development of the mineralised nanostructure in the intramembranously ossifying scapulae (shoulder bone). Using position-resolved scanning small angle X-ray scattering (SAXS), we quantified the degree and direction of mineral nanocrystallite alignment over the width of the scapulae, from the load bearing lateral border (LB) regions to the intermediate infraspinous fossa (IF) tissue. These measurements revealed a significant (p<0.05) increase in mineral nanocrystallite alignment in the LB when compared to the IF region, with increased tissue maturation in wild-type mice; this was absent in mice with rickets. The crystallites were more closely aligned to the macroscopic bone boundary in the LB when compared to the IF region in both wild type and Hpr mice, but the degree of alignment was reduced in Hpr mice. These findings are consistent with a correlation between the nanocrystallites within fibrils and in vivo muscular forces. Thus our results indicate a relevant mechanism for the observed increased macroscopic deformability in rickets, via a significant alteration in the mineral particle alignment, which is mediated by an altered spatial distribution of muscle forces.