P W. Dondl, J. Zimmer/J. Mech. Phys. Solids 52(2004)2057-2077 Table 4 Choice of functions to fit the elastic moduli =8252+16175+4753 Orthorhombic 1)2 +7.713,10°(P2-00002+40p3 165,105(p1-0.01)p2-0.000) dm=22291-00534)2+43470(2-000232)2 +3237(3-00256)2 +6.942(P1-0.0534)P3 2027(P2-000232)(P3 Large strains gs=150p1-00267)4+500p2-00012)2 in the interior of one box. On those three boxes,φ is defined asφ,φandφm, respectively. These functions are listed in Table 4; they are fitted in a straightforward manner to the elastic moduli given in Table 2 Second, p is extrapolated appropriately, here by a finite-element approach. We solve the biharmonic equation on the finite-element space spanned by 13 13x9 continuously differentiable rectangular box eleme depicted in Fi The element boundaries coincide with the breaks of Table 3. These elements are three-dimensional tensor prod ucts of one-dimensional C cubic Hermite-interpolation elements. Thus, they are a three-dimensional version of the Bogner-Fox-Schmitt element(Braess, 2001, Chap- ter Il, 5.10). All degrees of freedom of the three boxes containing the minima are prescribed to ensure that the solution respects the data of Table 4. a boundary of the finite-element domain is introduced, and the boundary conditions are set such that there is aC transition to a function large defining p for large strains(see Table 4) Clearly, there is freedom in the choice of the boundary, but as soon as it is sufficiently far away from the minima, this choice will become immaterial for the simulations. We could control the energy barriers between the minima as well. The method would be to add a force term to the biharmonic equation further to increase the function values on the nonprescribed degrees of freedom. In particular, the energy barrier is not deter mined via analyticity, such as for a polynomial ansatz of a given degree. Rather, the energy barrier can now be adjusted according to physical measurements(of ab initio calculations, say ) To demonstrate this, compare the energy barriers for the energy defined here, where no forcing term is used, with those of a lowest order polynomial energy(Fig. 4). The low-energy barrier of the polynomial ansatz sometimes makes the correct resolution of stable phases impossible for computational investigations. See the finite-element simulations in Section 4 We obtain a C smooth energy function. We will not smooth it, as it enters the quation of motion(3)(Section 4) only in the weak form. We mention some fine points in the procedure described above. To deal with the onlinear nature of the orbit space the breaks and hence the element matrices areP.W. Dondl, J. Zimmer / J. Mech. Phys. Solids 52 (2004) 2057 – 2077 2067 Table 4 Choice of functions to 3t the elastic moduli Tetragonal t = 8:252 1 + 161:752 + 47:53 Orthorhombic o = 175(1 − 0:01)2 +7:713 · 106(2 − 0:0001)2 + 403 −1:6675 · 105(1 − 0:01)(2 − 0:0001) Monoclinic m = 222:9(1 − 0:0534)2 + 43470(2 − 0:00232)2 +323:7(3 − 0:0256)2 −4867(1 − 0:0534)(2 − 0:00232) +6:942(1 − 0:0534)(3 − 0:0256) −20:27(2 − 0:00232)(3 − 0:0256) Large Strains large = 150(1 − 0:0267)4 + 500(2 − 0:0012)2 +50(3 − 0:0128)2 + 0:13 in the interior of one box. On those three boxes,  is de3ned as t, o and m, respectively. These functions are listed in Table 4; they are 3tted in a straightforward manner to the elastic moduli given in Table 2. Second,  is extrapolated appropriately, here by a 3nite-element approach. We solve the biharmonic equation on the 3nite-element space spanned by 13×13×9 continuously diIerentiable rectangular box elements, as depicted in Fig. 3. The element boundaries coincide with the breaks of Table 3. These elements are three-dimensional tensor prod￾ucts of one-dimensional C1 cubic Hermite-interpolation elements. Thus, they are a three-dimensional version of the Bogner–Fox–Schmitt element (Braess, 2001, Chap￾ter II, 5.10). All degrees of freedom of the three boxes containing the minima are prescribed to ensure that the solution respects the data of Table 4. A boundary of the 3nite-element domain is introduced, and the boundary conditions are set such that there is a C1 transition to a function large de3ning  for large strains (see Table 4). Clearly, there is freedom in the choice of the boundary, but as soon as it is suFciently far away from the minima, this choice will become immaterial for the simulations. We could control the energy barriers between the minima as well. The method would be to add a force term to the biharmonic equation further to increase the function values on the nonprescribed degrees of freedom. In particular, the energy barrier is not deter￾mined via analyticity, such as for a polynomial ansatz of a given degree. Rather, the energy barrier can now be adjusted according to physical measurements (of ab initio calculations, say). To demonstrate this, compare the energy barriers for the energy de3ned here, where no forcing term is used, with those of a lowest order polynomial energy (Fig. 4). The low-energy barrier of the polynomial ansatz sometimes makes the correct resolution of stable phases impossible for computational investigations. See the 3nite-element simulations in Section 4. We obtain a C1 smooth energy function. We will not smooth it, as it enters the equation of motion (3) (Section 4) only in the weak form. We mention some 3ne points in the procedure described above. To deal with the nonlinear nature of the orbit space, the breaks and hence the element matrices are