arXiv subjects

Lucas Bouck

Publications and source records attributed to Lucas Bouck.

8 recordsLinked to original sources

arXiv2026

Unconstrained Scheme for Geometrically Constrained Gradient Flows

In this paper, we study the approximation of gradient flows of harmonic maps, which serve as model problems for applications in micromagnetics, liquid crystals, and nonlinear plate bending. Harmonic maps are vector fields that are critical points of the Dirichlet energy subject to the constraint that the vector field be unit length pointwise. Most existing time-stepping schemes for gradient flows deal with the constraint by linearizing the unit length constraint at every step, which involves solving for the solution increment in the tangent space of the constraint. These schemes lead to robust control over the violation of the constraint, but require solving degenerate saddle point systems at every step that may be difficult to precondition. In this paper, we propose a scheme that first computes the unconstrained increment and then projects this increment pointwise onto the tangent space. With an additional stabilization, this scheme is energy stable under mild step size restrictions and provides robust control of the unit length constraint violation. Our new scheme only requires the solution of decoupled symmetric positive definite systems at every step, which translates to a large increase in computational efficiency. We also propose a computable a posteriori criterion and a variable time-stepping procedure that guarantee the stability of the scheme. We conclude with computational examples demonstrating the efficacy of the scheme, and present a computational extension of the scheme to nonlinear plate bending.

math.NA↗

arXiv2024

Commutativity and non-commutativity of limits in the nonlinear bending theory for prestrained microheterogeneous plates

In this paper we study the derivation of nonlinear bending models for prestrained elastic plates from three-dimensional non-linear elasticity via homogenization and dimension reduction. We compare effective models obtained by either simultaneously or consecutively passing to the $\Gamma$-limits as the thickness $h\ll1$ and the size of the material microstructure $\e\ll1$ vanish. In the regime $\e\ll h$ we show that the consecutive and simultaneous limit are equivalent, and also analyze the rate of convergence. In contrast, we observe that there are several different limit models in the case $h\ll \e$.

math.AP↗

arXiv2024

Plate theory for metric-constrained actuation of liquid crystal elastomer sheets

Liquid crystal elastomers (LCEs) marry the large deformation response of a cross-linked polymer network with the nematic order of liquid crystals pendent to the network. Of particular interest is the actuation of LCE sheets where the nematic order, modeled by a unit vector called the director, is specified heterogeneously in the plane of the sheet. Heating such a sheet leads to a large spontaneous deformation, coupled to the director design through a metric constraint that is now well-established by the literature. Here we go beyond the metric constraint and identify the full plate theory that underlies this phenomenon. Starting from a widely used bulk model for LCEs, we derive a plate theory for the pure bending deformations of patterned LCE sheets in the limit that the sheet thickness tends to zero using the framework of $\Gamma$-convergence. Specifically, after dividing the bulk energy by the cube of the thickness to set a bending scale, we show that all limiting midplane deformations with bounded energy at this scale satisfy the aforementioned metric constraint. We then identify the energy of our plate theory as an ansatz-free lower bound of the limit of the scaled bulk energy, and construct a recovery sequence that achieves this plate energy for all smooth enough midplane deformations. We conclude by applying our plate theory to a variety of examples.

cond-mat.soft↗

arXiv2024

Inf-Sup Stability of Parabolic TraceFEM

We develop a parabolic inf-sup theory for a modified TraceFEM semi-discretization in space of the heat equation posed on a stationary surface embedded in $\mathbb{R}^n$. We consider the normal derivative volume stabilization and add an $L^2$-type stabilization to the time derivative. We assume that the representation of and the integration over the surface are exact, however, all our results are independent of how the surface cuts the bulk mesh. For any mesh for which the method is well-defined, we establish necessary and sufficient conditions for inf-sup stability of the proposed TraceFEM in terms of $H^1$-stability of a stabilized $L^2$-projection and of an inverse inequality constant that accounts for the lack of conformity of TraceFEM. Furthermore, we prove that the latter two quantities are bounded uniformly for a sequence of shape-regular and quasi-uniform bulk meshes. We derive several consequences of uniform discrete inf-sup stability, namely uniform well-posedness, discrete maximal parabolic regularity, parabolic quasi-best approximation, convergence to minimal regularity solutions, and optimal order-regularity energy and $L^2 L^2$ error estimates. We show that the additional stabilization of the time derivative restores optimal conditioning of time-discrete TraceFEM typical of fitted discretizations.

math.NA↗

arXiv2024

Projection-Free Method for the Full Frank-Oseen Model of Liquid Crystals

Liquid crystals are materials that experience an intermediate phase where the material can flow like a liquid, but the molecules maintain an orientation order. The Frank-Oseen model is a continuum model of a liquid crystal. The model represents the liquid crystal orientation as a vector field and posits that the vector field minimizes some elastic energy subject to a pointwise unit length constraint, which is a nonconvex constraint. Previous numerical methods in the literature assumed restrictions on the physical constants or had regularity assumptions that ruled out point defects, which are important physical phenomena to model. We present a finite element discretization of the full Frank-Oseen model and a projection free gradient flow algorithm for the discrete problem in the spirit of Bartels (2016). We prove Gamma-convergence of the discrete to the continuous problem: weak convergence of subsequences of discrete minimizers and convergence of energies. We also prove that the gradient flow algorithm has a desirable energy decrease property. Our analysis only requires that the physical constants are positive, which presents challenges due to the additional nonlinearities from the elastic energy.

math.NA↗

arXiv2023

A hydrodynamical model of nematic liquid crystal films with a general state of orientational order

We develop a Q-tensor model of nematic liquid crystals occupying a stationary surface which represents a fluidic material film in space. In addition to the evolution due to Landau--de\,Gennes energy the model includes a tangent viscous incompressible flow along the surface. A thermodynamically consistent coupling of a two-dimensional flow and a three-dimensional Q-tensor dynamics is derived from the generalized Onsager principle following the Beris--Edwards system known in the flat case. The main novelty of the model is that it allows for a flow of an arbitrarily oriented liquid crystal so the Q-tensor is not anchored to the tangent plane of the surface, and also obeys an energy law. Several numerical experiments explore kinematical and dynamical properties of the novel model.

physics.flu-dyn↗

arXiv2023

Convergent FEM for a membrane model of liquid crystal polymer networks

We design a finite element method (FEM) for a membrane model of liquid crystal polymer networks (LCNs). This model consists of a minimization problem of a non-convex stretching energy. We discuss properties of this energy functional such as lack of weak lower semicontinuity. We devise a discretization with regularization, propose a novel iterative scheme to solve the non-convex discrete minimization problem, and prove stability of the scheme and convergence of discrete minimizers. We present numerical simulations to illustrate convergence properties of our algorithm and features of the model.

math.NA↗

arXiv2022

Reduced Membrane Model for Liquid Crystal Polymer Networks: Asymptotics and Computation

We examine a reduced membrane model of liquid crystal polymer networks (LCNs) via asymptotics and computation. This model requires solving a minimization problem for a non-convex stretching energy. We show a formal asymptotic derivation of the 2D membrane model from 3D rubber elasticity. We construct approximate solutions with point defects. We design a finite element method with regularization, and propose a nonlinear gradient flow with Newton inner iteration to solve the non-convex discrete minimization problem. We present numerical simulations of practical interests to illustrate the ability of the model and our method to capture rich physical phenomena.

math.NA↗