Narrow Results

We begin the mathematical study of Isogeometric Analysis based on NURBS (non-uniform rational B-splines). Isogeometric Analysis is a generalization of classical Finite Element Analysis (FEA) which possesses improved properties. For example, NURBS are capable of more precise geometric representation of complex objects and, in particular, can exactly represent many commonly engineered shapes, such as cylinders, spheres and tori. Isogeometric Analysis also simplifies mesh refinement because the geometry is fixed at the coarsest level of refinement and is unchanged throughout the refinement process. This eliminates geometrical errors and the necessity of linking the refinement procedure to a CAD representation of the geometry, as in classical FEA. In this work we study approximation and stability properties in the context of h-refinement. We develop approximation estimates based on a new Bramble–Hilbert lemma in so-called "bent" Sobolev spaces appropriate for NURBS approximations and establish inverse estimates similar to those for finite elements. We apply the theoretical results to several cases of interest including elasticity, isotropic incompressible elasticity and Stokes flow, and advection-diffusion, and perform numerical tests which corroborate the mathematical results. We also perform numerical calculations that involve hypotheses outside our theory and these suggest that there are many other interesting mathematical properties of Isogeometric Analysis yet to be proved.

This paper analyzes the numerical solution of a class of nonlinear Schrödinger equations by Galerkin finite elements in space and a mass and energy conserving variant of the Crank–Nicolson method due to Sanz-Serna in time. The novel aspects of the analysis are the incorporation of rough, discontinuous potentials in the context of weak and strong disorder, the consideration of some general class of nonlinearities, and the proof of convergence with rates in $L^{\infty }(L^2)$ under moderate regularity assumptions that are compatible with discontinuous potentials. For sufficiently smooth potentials, the rates are optimal without any coupling condition between the time step size and the spatial mesh width.

In this paper we propose a new space and time finite elements discretization of the Landau–Lifshitz equations which may be readily used for numerical computations. We then prove its convergence to a weak solution in the sense given by Alouges and Soyeur or Labbé in the literature.

We provide an overview of the Arbitrary Lagrangian–Eulerian Variational Multiscale (ALE-VMS) and Space–Time Variational Multiscale (ST-VMS) methods we have developed for computer modeling of wind-turbine rotor aerodynamics and fluid–structure interaction (FSI). The related techniques described include weak enforcement of the essential boundary conditions, Kirchhoff–Love shell modeling of the rotor-blade structure, NURBS-based isogeometric analysis, and full FSI coupling. We present results from application of these methods to computer modeling of NREL 5MW and NREL Phase VI wind-turbine rotors at full scale, including comparison with experimental data.

A nonlocal Cahn–Hilliard model with a non-smooth potential of double-well obstacle type that promotes sharp interfaces in the solution is presented. To capture long-range interactions between particles, a nonlocal Ginzburg–Landau energy functional is defined which recovers the classical (local) model as the extent of nonlocal interactions vanish. In contrast to the local Cahn–Hilliard problem that always leads to diffuse interfaces, the proposed nonlocal model can lead to a strict separation into pure phases of the substance. Here, the lack of smoothness of the potential is essential to guarantee the aforementioned sharp-interface property. Mathematically, this introduces additional inequality constraints that, in a weak formulation, lead to a coupled system of variational inequalities which at each time instance can be restated as a constrained optimization problem. We prove the well-posedness and regularity of the semi-discrete and continuous in time weak solutions, and derive the conditions under which pure phases are admitted. Moreover, we develop discretizations of the problem based on finite element methods and implicit–explicit time-stepping methods that can be realized efficiently. Finally, we illustrate our theoretical findings through several numerical experiments in one and two spatial dimensions that highlight the differences in features of local and nonlocal solutions and also the sharp interface properties of the nonlocal model.

A mathematical analysis of local and nonlocal phase-field models of tumor growth is presented that includes time-dependent Darcy–Forchheimer–Brinkman models of convective velocity fields and models of long-range cell interactions. A complete existence analysis is provided. In addition, a parameter-sensitivity analysis is described that quantifies the sensitivity of key quantities of interest to changes in parameter values. Two sensitivity analyses are examined; one employing statistical variances of model outputs and another employing the notion of active subspaces based on existing observational data. Remarkably, the two approaches yield very similar conclusions on sensitivity for certain quantities of interest. The work concludes with the presentation of numerical approximations of solutions of the governing equations and results of numerical experiments on tumor growth produced using finite element discretizations of the full tumor model for representative cases.

In this paper, we study the thin-film limit of the micromagnetic energy functional in the presence of bulk Dzyaloshinskii–Moriya interaction (DMI). Our analysis includes both a stationary $\mathrm{\Gamma }$-convergence result for the micromagnetic energy, as well as the identification of the asymptotic behavior of the associated Landau–Lifshitz–Gilbert equation. In particular, we prove that, in the limiting model, part of the DMI term behaves like the projection of the magnetic moment onto the normal to the film, contributing this way to an increase in the shape anisotropy arising from the magnetostatic self-energy. Finally, we discuss a convergent finite element approach for the approximation of the time-dependent case and use it to numerically compare the original three-dimensional (3D) model with the 2D thin-film limit.

This paper presents a parallel solver for the Monodomain and Bidomain systems modeling the bioelectrical activity of cardiac tissue in three dimensions. These systems consists of nonlinear parabolic reaction-diffusion equations coupled with a stiff system of several ordinary differential equations, known as Luo-Rudy I model. Instead of the more common explicit or semi-implicit time discretizations, we consider here the fully implicit Backward Euler and linearly implicit Rosenbrock methods, that allow a time step-size selection without stability constraints. Trilinear finite elements are employed in the space discretization. The nonlinear systems originated at each time step of Backward Euler are solved by a Newton-Krylov method, where the outer iteration of an inexact Newton method solves the linear systems of the Jacobian update with a Krylov method such as GMRES with a block Jacobi preconditioner. The same linear solver is applied to the linear systems of the Rosenbrock method. Parallel numerical results on a Linux Cluster show that the proposed parallel solver is scalable for the Monodomain system, with both linear and nonlinear iteration counts bounded from above by a constant independent of the number of processors. This does not hold for the Bidomain system, since the linear iteration counts increase with the number of processors.

In this paper we present a new approach in the study of Aorto–Coronaric bypass anastomoses configurations. The theory of optimal control based on adjoint formulation is applied in order to optimize the shape of the zone of the incoming branch of the bypass (the toe) into the coronary. The aim is to provide design indications in the perspective of future development for prosthetic bypasses. With a reduced model based on Stokes equations and a vorticity functional in the down field zone of bypass, a Taylor-like patch is found. A feedback procedure with Navier–Stokes fluid model is proposed based on the analysis of wall shear stress and its related indexes such as OSI.

In this work, a parallel three-dimensional solver for numerical simulations in computational electrocardiology is introduced and studied. The solver is based on the anisotropic Bidomain cardiac model, consisting of a system of two degenerate parabolic reaction–diffusion equations describing the intra and extracellular potentials of the myocardial tissue. This model includes intramural fiber rotation and anisotropic conductivity coefficients that can be fully orthotropic or axially symmetric around the fiber direction. The solver also includes the simpler anisotropic Monodomain model, consisting of only one reaction–diffusion equation. These cardiac models are coupled with a membrane model for the ionic currents, consisting of a system of ordinary differential equations that can vary from the simple FitzHugh–Nagumo (FHN) model to the more complex phase-I Luo–Rudy model (LR1). The solver employs structured isoparametric Q₁ finite elements in space and a semi-implicit adaptive method in time. Parallelization and portability are based on the PETSc parallel library. Large-scale computations with up to O(10⁷) unknowns have been run on parallel computers, simulating excitation and repolarization phenomena in three-dimensional domains.

We present and analyze new multi-species phase-field mathematical models of tumor growth and ECM invasion. The local and nonlocal mathematical models describe the evolution of volume fractions of tumor cells, viable cells (proliferative and hypoxic cells), necrotic cells, and the evolution of matrix-degenerative enzyme (MDE) and extracellular matrix (ECM), together with chemotaxis, haptotaxis, apoptosis, nutrient distribution, and cell-to-matrix adhesion. We provide a rigorous proof of the existence of solutions of the coupled system with gradient-based and adhesion-based haptotaxis effects. In addition, we discuss finite element discretizations of the model, and we present the results of numerical experiments designed to show the relative importance and roles of various effects, including cell mobility, proliferation, necrosis, hypoxia, and nutrient concentration on the generation of MDEs and the degradation of the ECM.

The aim of this paper is to show that, for a linear second-order hyperbolic equation discretized by the backward Euler scheme in time and continuous piecewise affine finite elements in space, the adaptation of the time steps can be combined with spatial mesh adaptivity in an optimal way. We derive a priori and a posteriori error estimates which admit, as much as it is possible, the decoupling of the errors committed in the temporal and spatial discretizations.

The paper describes a finite element investigation into the buckling, under uniform external pressure, of four submarine pressure hulls. Two of these hulls were traditional ring-stiffened types, but two of these hulls were in the form of corrugated circular cylinders. The latter design was based on an invention by the present author. The investigation found that the new design was structurally efficient and in the case of the smaller vessel, it was found to be structurally more efficient than the conventional design.

Another investigation, based on axisymmetric plastic buckling, was conducted on the two corrugated vessels, to determine if they were prone to collapse through the bellows' mode of failure. This investigation was carried out because former critics of this work stated that the corrugated circular cylinders would fail by the bellows' mode of failure. Neither of the two corrugated pressure hulls was found to fail through the bellows' mode of failure, thereby completely vindicating the present author. The author, however, concludes that the bellows' mode of failure can occur if the cone angles were too large.

Two of the finite element theories were based on the author's work, but the giant computer package ANSYS was also used to study non-symmetric bifurcation buckling. This work was carried out to vindicate the author's in-house computer programs, which were simpler to use than ANSYS.

This paper deals with a posteriori error estimators for the linear finite element approximation of second-order elliptic eigenvalue problems in two or three dimensions. First, we give a simple proof of the equivalence, up to higher order terms, between the error and a residual type error estimator. Second, we prove that the volumetric part of the residual is dominated by a constant times the edge or face residuals, again up to higher order terms. This result was not known for eigenvalue problems.

We describe all operators on scalar finite element spaces which appear as the restriction of a second-order (linear) differential operator. More precisely we provide a family of isomorphisms between this space of discrete differential operators and products of some exotic finite element spaces. This provides a unified framework for the Regge calculus of numerical relativity and the Nédélec edge elements of computational electromagnetism.

A Balancing Domain Decomposition by Constraints (BDDC) preconditioner for Isogeometric Analysis of scalar elliptic problems is constructed and analyzed by introducing appropriate discrete norms. A main result of this work is the proof that the proposed isogeometric BDDC preconditioner is scalable in the number of subdomains and quasi-optimal in the ratio of subdomain and element sizes. Another main result is the numerical validation of the theoretical convergence rate estimates by carrying out several two- and three-dimensional tests on serial and parallel computers. These numerical experiments also illustrate the preconditioner performance with respect to the polynomial degree and the regularity of the NURBS basis functions, as well as its robustness with respect to discontinuities of the coefficient of the elliptic problem across subdomain boundaries.

We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian ${(-\mathrm{\Delta })}^s$ in a Lipschitz bounded domain $\mathrm{\Omega }\subset {ℝ}^n$ satisfying the exterior ball condition. The weight is a power of the distance to the boundary $\partial \mathrm{\Omega }$ of $\mathrm{\Omega }$ that accounts for the singular boundary behavior of the solution for any $0<s<1$. These bounds then serve us as a guide in the design and analysis of a finite element scheme over graded meshes for any dimension $n$, which is optimal for $n=2$.

We consider the strong solution of the 2D Navier–Stokes equations in a torus subject to an additive noise. We implement a fully implicit time numerical scheme and a finite element method in space. We prove that the space-time rate of convergence is the “optimal” one, namely, $\eta \in [0,1/2)$ in time and 1 in space. Let us mention that the coefficient $\eta$ is equal to the time regularity of the solution with values in ${𝕃}^2$. Our method relies on the existence of finite exponential moments for both the solution and its time approximation. Unlike previous results, our main new idea is the use of a discrete Grönwall lemma for the error estimate without any localization.

This paper presents a computational approach for composite materials subjected to ballistic impact. The composite materials are composed of layers of fabric embedded in a matrix material. The fabric is represented by bar elements and the matrix material is represented by solid elements. The geometric characteristics of these composite materials are three dimensional, which in turn requires a three dimensional analysis. The strength and failure characteristics of the fabric are determined directly from the actual fiber characteristics. For the matrix material it is necessary to modify the solid elements, because much of the volume of the solid elements is occupied by the volume of the fabric. The resulting modifications include the determination of effective density, stiffness, strength and failure characteristics for the solid elements. This approach has been shown to provide good agreement with experimental ballistics data. Because this approach treats each of the component materials individually (instead of blending them into a homogeneous anisotropic material), it is possible to examine the effects of individual material properties for both the fabric and the matrix material. This includes the capability to allow for failure of the fabric material and/or the matrix material. Parametric computational results are included to illustrate some of these effects.

Given a cellular complex, we construct spaces of differential forms which form a complex under the exterior derivative, which is isomorphic to the cochain complex of the cellular complex. The construction applies in particular to subsets of Euclidean space divided into polyhedra, for which it provides, for each k, a space of k-forms with a basis indexed by the set of k-dimensional cells. In the framework of mimetic finite differences, the construction provides a conforming reconstruction operator. The construction requires auxiliary spaces of differential forms on each cell, for which we provide two examples. When the cells are simplexes, the construction can be used to recover the standard mixed finite element spaces also called Whitney forms. We can also recover the dual finite elements previously constructed by A. Buffa and the author on the barycentric refinement of a two-dimensional mesh.