We propose a mathematical framework to effectively study lattice materials with periodic and non-periodic structures over entire spaces in one, two, and three dimensions. The existence and uniqueness of solutions for periodic lattice problems with absolute terms are proved in discrete Sobolev spaces. By Fourier transform discrete lattice problems are converted to semi-discrete problems for which similar results are established in semi-discrete Sobolev spaces. For lattice problems without absolute terms, additional conditions are imposed on data for the existence and uniqueness of solutions in discrete energy spaces in one, two and three dimensions. Two concrete examples are analyzed in the proposed mathematical framework. The mathematical framework, methodology and techniques in this paper can be utilized or generalized to non-periodic lattices on entire spaces and boundary value problems on lattices.

In the finite element method, a standard approach to mesh tying is to apply Lagrange multipliers. If the interface is curved, however, discretization generally leads to adjoining surfaces that do not coincide spatially. Straightforward Lagrange multiplier methods lead to discrete formulations failing a first-order patch test [12]. A least-squares method is presented here for mesh tying in the presence of gaps and overlaps. The least-squares formulation for transmission problems [5] is extended to settings where subdomain boundaries are not spatially coincident. The new method is consistent in the sense that it recovers exactly global polynomial solutions that are in the finite element space. As a result, the least-squares mesh tying method passes a patch test of the order of the finite element space by construction. This attractive computational property is illustrated by numerical experiments.

A number of practical engineering problems require the repeated simulation of unsteady fluid flows. These problems include the control, optimization and uncertainty quantification of fluid systems. To make many of these problems tractable, reduced-order modeling has been used to minimize the simulation requirements. For nonlinear, time-dependent problems, such as the Navier-Stokes equations, reduced-order models are typically based on the proper orthogonal decomposition (POD) combined with Galerkin projection. We study several modifications to this reduced-order modeling approach motivated by the optimization problem underlying POD. Our discussion centers on a method known as the principal interval decomposition (PID) due to IJzerman.

Determining accurate statistical information about outputs from ensembles of realizations is not generally possible whenever the input-output map involves the (computational) solution of systems of nonlinear partial differential equations (PDEs). This is due to the high cost of effecting each realization. Recently, in applications such as control and optimization that also require multiple solutions of PDEs, there has been much interest in reduced-order models (ROMs) that greatly reduce the cost of determining approximate solutions. We explore the use of ROMs for determining outputs that depend on solutions of stochastic PDEs. One is then able to cheaply determine much larger ensembles, but this increase in sample size is countered by the lower fidelity of the ROM used to approximate the state. In the contexts of proper orthogonal decomposition-based ROMs, we explore these counteracting effects on the accuracy of statistical information about outputs determined from ensembles of solutions.

In this paper, we investigate an efficient numerical method to identify an optimal impedance factor for mitigating radiated engine noise. The engine tone-noise wavenumber is treated as a random variable. We prove the existence of the sensitivity derivative of the state variable (which is the acoustic pressure) with respect to the random wavenumber. The proposed numerical method is based on the stratified Monte Carlo algorithm whose convergence is accelerated by exploiting the sensitivity derivative information.

In this paper we analyze several first-order systems of Oseen-type equations that are obtained from the time-dependent incompressible Navier-Stokes equations after introducing the additional vorticity and possibly total pressure variables, time-discretizing the time derivative and linearizing the non-linear terms. We apply the [$L^2$, $L^2$, $L^2$] least-squares finite element scheme to approximate the solutions of these Oseen-type equations assuming homogeneous velocity boundary conditions. All of the associated least-squares energy functionals are defined to be the sum of squared $L^2$ norms of the residual equations over an appropriate products space. We first prove that the homogeneous least-squares functionals are coercive in the $H^1 \times L^2 \times L^2$ norm for the velocity, vorticity, and pressure, but only continuous in the $H^1 \times H^1 \times H^1$ norm for these variables. Although equivalence between the homogeneous least-squares functionals and one of the above two product norms is not achieved, by using these a priori estimates and additional finite element analysis we are nevertheless able to prove that the least-squares method produces an optimal rate of convergence in the $H^1$ norm for velocity and suboptimal rate of convergence in the $L^2$ norm for vorticity and pressure. Numerical experiments with various Reynolds numbers that support the theoretical error estimates are presented. In addition, numerical solutions to the time-dependent incompressible Navier-Stokes problem are given to demonstrate the accuracy of the semi-discrete [$L^2, L^2, L^2$] least-squares finite element approach.

We consider the Stokes-Oldroyd equations, defined here as the Stokes equations with the Newtonian constitutive equation explicitly included. Thus a polymer-like stress tensor is included so that the dependent variable structure of a viscoelastic model is in place. The energy equation is coupled with the mass, momentum, and constitutive equations through the use of temperature-dependent viscosity terms in both the constitutive model and the momentum equation. Earlier works assumed temperature-dependent constitutive (polymer) and Newtonian (solvent) viscosities when describing the model equations, but made the simplifying assumption of a constant solvent viscosity when carrying out analysis and computations; we assume no such simplification. Our analysis coupled with numerical solution of the problem with both temperature-dependent viscosities distinguishes this work from earlier efforts.

We study numerical approximations of a recently proposed phase field model for the vesicle membrane deformations governed by the variation of the elastic bending energy. Both the spatial discretization for the equilibrium problem with given volume and surface area constraints and the time discretization of a dynamic problem via gradient flow are considered. Convergence results of the numerical approximations are proved.

We study the scaling properties of heterogeneities in nematic (liquid crystal) polymers that are generated by pressure-driven, capillary Poiseuille flow. These studies complement our earlier drag-driven structure simulations and analyses. We use the mesoscopic Doi-Marrucci-Greco model, which incorporates excluded-volume interactions of the rod-like particle ensemble, distortional elasticity of the dispersion, and hydrodynamic feedback through orientation dependent viscoelastic stresses. The geometry likewise introduces anchoring conditions on the nano-rods which touch the solid boundaries. We first derive flow-orientation steady-state structures for three different anchoring conditions, by asymptotic analysis in the limit of weak pressure gradient. These closed-form expressions yield scaling laws, which predict how lengthscales of distortions in the flow and orientational distributions vary with strength of the excluded volume potential, molecule geometry, and distortional elasticity constants. Next, the asymptotic structures are verified by direct numerical simulations, which provide a high level benchmark on the numerical code and algorithm. Finally, we calculate the effective (thermal or electrical) conductivity tensor of the composite films, and determine scaling behavior of the effective property enhancements generated by capillary Poiseuille flow.

We present a numerical algorithm for nano-rod suspension flows, and provide benchmark simulations of a plane Couette cell experiment. The system consists of a Smoluchowski equation for the orientational distribution function of the nano-rods together with the Navier-Stokes equation for the solvent with an orientation-dependent stress. The rigid rods interact through nonlocal excluded-volume and distortional elasticity potentials and hydrodynamic interactions. The algorithm resolves full orientational configuration space (a spherical harmonic Galerkin expansion), two dimensional physical space (method of lines discretization), and time (spectral deferred corrections), and employs a velocity-pressure formulation of the Navier-Stokes equation. This method extends our previous solver [25] from 1D to 2D in physical space.

In this paper we established a Carleman estimate for the elasticity system with the residual stresses. As an application of this estimate we obtain exact controllability results for the same system with locally distributed control.

In this paper we describe a one-dimensional interface problem for the heat equation, with a nonlinear (quadratic) jump condition at the interface. We derive a numerical method for approximating solutions of this nonlinear problem and provide some results from numerical experiments. The novelty of this problem is precisely this nonlinear (quadratic) jump condition, and it arises in the study of polymeric ion-selective electrodes and ion sensors.

Although the methodology of centroidal Voronoi tessellation (CVT) has been widely used for mesh generation on complex geometries, a clear characterization of the influence of geometric constraints on the CVT-based meshing is still lacking. In this paper, we first give a clear definition of the conforming centroidal Voronoi Delaunay triangulation (CCVDT) and then propose an efficient algorithm for its construction in two dimensional space. Finally, we show the high-quality of CCVDT meshes and the effectiveness and robustness of our algorithm through extensive examples.

We consider the following problem: Given a finite number of sails of different sizes with prescribed shapes, given the speed of the wind, the weight of a surfer, the size and shape of a surfboard and a corresponding fin; determine the sail and the course such that the windsurfer achieves maximal speed. We develop a simple model for describing the movement of the windsurfer for maximizing the speed in terms of a nonlinear ODE (ordinary differential equation). We conclude with the presentation of corresponding numerical results, some remarks on their validation and more involved models.

In this paper, we study a reduced-order modeling for Burgers equations. Review of the CVT (centroidal Voronoi tessellation) approaches to reduced-order bases are provided. In CVT-reduced order modelling, we start with a snapshot set just as is done in a POD (Proper Orthogonal Decomposition)-based setting. We shall investigate the technique of CVT as a procedure to determine the basis elements for the approximating subspaces. Some numerical experiments including comparison of CVT-based algorithm with numerical results obtained from FEM (finite element method) and POD-based algorithm are reported. Finally, we apply CVT-based reduced order modeling technique to a feedback control problem for Burgers equation.

The matching velocity problem for the steady-state Navier-Stokes system is considered. We introduce an extended domain method for solving optimal boundary control problems. The Lagrangian multiplier method is applied to the extended domain with distributed controls and used to determine the optimality system and the control over the boundary of the inner domain. The existence, the differentiability and the optimality system of the control problem are discussed. With this method inflow controls are shown to be numerical reliable over a large admissible control set. Numerical tests for steady-state solutions are presented to prove the effectiveness and robustness of the method for flow matching.

In this paper, we consider the stabilization of steady state solutions to Navier-Stokes equations by boundary feedback control. The feedback control is determined by solving a linear quadratic regulator problem associated with the linearized Navier-Stokes equations. The control is effected through suction and blowing at the boundary. We show that the linear feedback control provides global exponential stabilization of the steady state solutions to the Navier-Stokes equations for arbitrary Reynolds number. This feedback is shown to provide global stability in both $L^2$ and $H^1$-norms.

Numerical solutions of exact controllability problems for linear and semilinear 2-d wave equations with distributed controls are studied. Exact controllability problems can be solved by the corresponding optimal control problems. The optimal control problem is reformulated as a system of equations (an optimality system) that consists of an initial value problem for the underlying (linear or semilinear) wave equation and a terminal value problem for the adjoint wave equation. The discretized optimality system is solved by a shooting method. The convergence properties of the numerical shooting method in the context of exact controllability are illustrated through computational experiments.

We study the numerical errors in finite elements discretizations of a time relaxation model of fluid motion:
                          $u_t + u\cdot \nabla u + \nabla p - \nu\Delta u + \chi u^* = f$ and $\nabla \cdot u = 0$
In this model, introduced by Stolz, Adams, and Kleiser, $u^*$ is a generalized fluctuation and $\chi$ the time relaxation parameter. The goal of inclusion of the $\chi u^*$ is to drive unresolved fluctuations to zero exponentially. We study convergence of discretization of model to the model's solution as $h$, $\Delta t \rightarrow 0$. Next we complement this with an experimental study of the effect the time relaxation term (and a nonlinear extension of it) has on the large scales of a flow near a transitional point. We close by showing that the time relaxation term does not alter shock speeds in the inviscid, compressible case, giving analytical confirmation of a result of Stolz, Adams, and Kleiser.

Pulsed laser materials processing incorporates many physical processes, the modeling of which requires disparate techniques. We focus on two closely related topics that arise in laser processing. First, we illustrate by use of numerical simulation the distinction between metals and ceramics in their response to laser fluence. Second we explain recent experiments that indicate that the substrate heater is a strong control parameter in laser processing, and this can be understood from the point of view of shock dynamics.

A discontinuous Galerkin finite element method for optimal control problems having states constrained by linear parabolic PDE's is examined. The spacial operator may depend on time and need not be self-adjoint. The schemes considered here are discontinuous in time but conforming in space. Fully-discrete error estimates of arbitrary order are presented and various constants are tracked. In particular, the estimates are valid for small values of $\alpha$, $\gamma$, where $\alpha$ denotes the penalty parameter of the cost functional and $\gamma$  the coercivity constant. Finally, error estimates for the convection dominated convection-diffusion equation are presented, based on a Lagrangian moving mesh approach.

A terminal-state optimal control problem for semilinear parabolic equations is studied in this paper. The control objective is to track a desired terminal state and the control is of the distributed type. A distinctive feature of this work is that the controlled state and the target state are allowed to have nonmatching boundary conditions. The existence of an optimal control solution is proved. We also show that the optimal solution depending on a parameter $\gamma$ gives solutions to the approximate controllability problem as $\gamma \rightarrow 0$. Error estimates are obtained for semidiscrete (spatially discrete) approximations of the optimal control problem. A gradient algorithm is discussed and numerical results are presented.

It is shown how an arbitrary set of points in the hypercube can be Latinized, i.e., can be transformed into a point set that has the Latin hypercube property. The effect of Latinization on the star discrepancy and other uniformity measures of a point set is analyzed. For a few selected but representative point sampling methods, evidence is provided to show that Latinization lowers the star discrepancy measure. A novel point sampling method is presented based on centroidal Voronoi tessellations of the hypercube. These point sets have excellent volumetric distributions, but have poor star discrepancies. Evidence is given that the Latinization of CVT points sets greatly lowers their star discrepancy measure but still preserves superior volumetric uniformity. As a result, means for determining improved Latin hypercube point samples are given.