A numerical method is established to solve the problem of minimizing a nonquasiconvex potential energy. Convergence of the method is proved both in the case on its own and in the case when it is combined with a weak boundary condition. Numerical examples are given to show that the method, especially when applied together with a continuation method and some other numerical techniques, is not only successful and e

Some micromagnetic phenomena can be modelled by a minimization problem of a nonconvex energy. A numerical method to compute the micromagnetic field, which gives rise to a finite dimensional unconstrained minimization problem, is given and analyzed. In our method, the Maxwell's equation defined on the whole space is solved by a finite element method using artificial boundary, and the highly oscillatory magnetization structure is approximated by an element-wise constant Young measure supported on a finite number of unknown points on the unit sphere. Numerical experiments on some uniaxial and cubic anisotropic energy densities show that the method is e

The mesh transformation method is applied on a two dimensional elastic crystal model to study the formation of laminated microstructure in austenitemartensite phase transition when certain external loads are applied. Numerical experiments show that simple laminated microstructures with various volume fractions and twin width can be obtained by varying the loads. Numerical experiments also show that second order laminated microstructure with branched peedle-like laminates can also be obtained by oertain loads.

A multiscale model and numerical method for computing microstructures with large and inhomogeneous deformation is established, in which the microscopic and macroscopic information is recovered by coupling the finite order rank-one convex envelope and the finite element method. The method is capable of computing microstructures which are locally finite order laminates. Numerical experiments on a double well problem show that plenty of stress free large deformations can be achieved by microstructures consisting of piecewise simple twin laminates.

We report some numerical results on the computation of needlelike microstructrues at mesoscopic scale obtained by applying the mesh transformation method, which basically includes a mesh optimization into the finite element approximations, on a nonquasiconvex elastic crystal model. Numerical experiments show that the branched needle-like microstructures can be well resolved near the interfaces between twinned layers of martensite and a single variant of martensite, which are in good qualitative agreement with the physical experiments.

A rotation transformation method and an incremental crystallization method are developed to overcome some of the diffculties involved in the computation of microstructures. The numerical method based on these techniques is proved to converge. To increase further the accuracy of the computation, a technique is applied to remove the boundary e

Variational problems with non-rank-one connected double well potential are considered. It is proved that the problem has a laminated microstructure solution which is uniquely determined by the two potential wells yet certainly not characterized by the fine oscillations between them because of the lack of rank-one connection. The laminated microstructure is explicitly worked out. It is also shown that an application of a nonconforming finite element method can cause over-relaxation and fail to approximate the right microstructure.

The problem of minimizing multiple integral functionals with nonquasiconvex integrands is considered. A numerical method, which is based on an alternative minimizing problem to the relaxed problem and thus uses no quasiconvex envelope of the integrands nor its numerical approximation in the computation, is introduced to approximate simultaneously the highly oscillating minimizing sequences, or in other words microstructures, and the minimizers of the corresponding relaxed problem. Existence and convergence of the discrete solutions are proved and an error estimate is obtained. A numerical example is given.

A mathematical model is given to study how the twin width of the needle-like laminate is related to the twin length and the surface energy density in austenite-martensite phase transition. A umerical method based on the mesh transformation method is given, and numerical experiments are made to establish such a relationship for an elastic crystal model.