This dissertation is concerned with the numerical analysis and computation of variationalmodels related to liquid crystals (LCs) and liquid crystal polymer networks (LCNs) as well as modeling of LCNs.

We first present a finite element method and projection free gradient flow to minimize theFrank-Oseen energy of nematic liquid crystals. The Frank-Oseen model is a continuum model that represents the liquid crystal with a vector field that must satisfy a nonconvex unit-length constraint pointwise. We prove convergence of minimizers of the discrete problem to minimizers of the continuous problem using the framework of Gamma-convergence. The convergence analysis has no restrictions on the elastic constants or regularity of the solution beyond that required for existence of minimizers. Due to the low regularity requirement, the method can capture point defects. We also propose a projection free gradient flow algorithm to compute critical points of the discrete energy. The gradient flow is conditionally stable under a mild restriction on the numerical parameters. We finally present computations illustrating the influence of the elastic constants on point defects as well as the influence of external magnetic fields.

The second part of this dissertation is concerned with modeling, numerical analysis, andcomputation of thin LCNs. We first begin from a classical 3D energy of LCN and use Kirchhoff- Love asymptotics to derive a reduced 2D membrane model. We then prove many properties of the membrane model including a pointwise metric condition that zero energy states must satisfy and construct a formal method to approximate configurations of LCN from higher degree defects that approximately match this pointwise condition. To conclude, we develop a finite element method to minimize the stretching energy. A key component of the discrete energy is to introduce a regularization that is inspired by a bending energy for LCN, which is also derived in this dissertation. We prove convergence of minimizers of the discrete problem to zero energy states of the continuous problem in the spirit of Gamma-convergence. To compute critical points of the discrete problem, we propose a fully implicit gradient flow with Newton sub-iteration and study its superlinear convergence under suitable assumptions. We finish with many simulations that highlight interesting features of LCNs, including configurations arising from LC defects and nonisometric origami.