On the Numerical Modeling of Deformations of Pressurized Martensitic Thin Films

Abstract

Abstract. We propose, analyze, and compare several numerical methods for the computation of the deformation of a pressurized martensitic thin film. Numerical results have been obtained for the hysteresis of the deformation as the film transforms reversibly from austenite to martensite. Mathematics Subject Classification.

Modeling the Behavior of Heat-Shrinkable Thin Films

“…Thus, the asymptotically derived bending energy (16) agrees formally with the bending energy (14) obtained by using -convergence techniques. It is noted in [21] that for the special case of an isotropic material, minimizing out v in (15) yields the -limiting functional of the form…”

“…To perform the quadrature exactly, we shall use the 7-point Gaussian quadrature rule described in [17] and exact for polynomials of degree 5. Finally, the energy shall be minimized by a variant of the Fletcher-Reeves conjugate gradient algorithm [24,43] that has been used successfully by one of the authors to minimize similar energy functionals in [8][9][10][11][12]14]. From the discussion in Sect.…”

Numerical analysis of a model for ferromagnetic shape memory thin films

“…where We used the Polak-Ribière nonlinear conjugate gradient (NCG) method to minimize the energy in the finite element spaceĀ τ,y 0 with the above deformation and magnetization as the initial state [6]. The NCG method converged to the local minimizer of the energy shown in Figure 3(b).…”

A finite element model for martensitic thin films

“…. , 1009 by computing a local minimum for the energyĒ 0 (y, b; θ , σ ) from the Polak-Ribière conjugate gradient method [10,38] with initial iterate…”

“…As in [10], we actually computed the gradients used in the conjugate gradient iterations by replacing the "mass" matrix on the left-hand side of (7.3) by the identity matrix with respect to the classical Lagrangian shape functions for the continuous, piecewise linear finite element space [17]. With this replacement, the mean-zero property S y = 0 is only approximately satisfied, but we can replace y by y − |S| −1 S y to regain the mean-zero property at any time in the iteration without affecting the computation since the energy (4.3) is invariant with respect to the translation y → y + c for c ∈ R 3 .…”