Fast MATLAB evaluation of nonlinear energies using FEM in 2D and 3D: Nodal elements

MATLAB implementation of hp finite elements on rectangles

“…where for p = 1: s is the index of a node i ∈ {1, 2, 3, 4}, for p ≥ 2: s is the index of an edge j ∈ {1, 2, 3, 4}, for p ≥ 4: s = 4 + β, where β is the local index of a bubble function (7).…”

“…The assemblies of FEM matrices are based on our vectorized codes [1,2]. The names of most of the mesh attributes and the domain rectangulation algorithms are taken from [7].…”

Vectorized MATLAB Implementation of the Incremental Minimization Principle for Rate-Independent Dissipative Solids Using FEM: A Constitutive Model of Shape Memory Alloys

“…For (hyper)elastic materials, D τ disappears, α reduces to the displacement field (u), and E τ involves only gradient(s) of u (cf. [11]). Through different mathematical forms, the dissipation functional allows one to describe different types of path-dependent material responses.…”

“…Recently, Moskovka and Valdman [11] developed and successfully verified the fully vectorized implementation of hyperelastic constitutive models into the finite element method (FEM) through mathematical optimization. Their approach relies on a concurrent (loop-free) evaluation of (both linear and gradient) energy contributions to the global energy functional and efficient evaluation of energy gradients (needed in optimization) by employing the concept of nodal patches.…”

“…As a representative example, we employ the previously developed constitutive model for the phase transformation and reorientation processes in shape memory alloys [22], which belongs to the class of models with strongly coupled rate-independent dissipation processes. This requires several extensions and modifications of the computational framework in [11]: (1) The set of thermodynamic descriptors of the system is enriched by internal variables (and temperature) fields. (2) Both the displacement field and fields of internal variables enter the minimization process.…”