Let us now concentrate on the finite element method. The fundamental equation to be solved is the equilibrium Eq. 11.1 (or the balance Eq. 11.6 for diffusion phenomena). As the numerical methods give an approximate solution, the equilib — rium/balance equation has to be solved with the best compromise. This is obtained by a global weak form of the local equation. Using weighted residuals, for solid mechanics, one obtains:
J [v, j Sel}]dV = J P, Sl, dV + j Р8Ш (11.17)
V V A
And for diffusion phenomena:
j [,SSp — f, d, (5p)]dV = j QSpdV + j qSpdA (11.18)
V V A
where p and q are surface terms of imposed loads/fluxes. The weighting functions are denoted 81 and 8p, and $£ represents a derivative of the weighting function based on the Cauchy’s strain derivate operator. An equivalent equation could be obtained based on the virtual power principle. The 81 and 8p terms would then be interpreted as virtual arbitrary displacements and pressures. Within the finite element method, the global equilibrium/balance equation will be verified for a number of fundamental cases equivalent to the degrees of freedom (d. o.f.) of the problem, i. e. the number of nodes times the number of degrees of freedom per node, minus the number of imposed values. The corresponding weighting functions will have simple forms based on the element shape functions.[26]
Giving a field of stress or of flux, using the weighting functions, one will obtain a value for each d. o.f., which is equivalent to a nodal expression of the equilib — rium/balance equation.
More precisely, for solid mechanics problems, one will obtain internal forces equivalent to stresses at each node, L:
Fu = f (JijBLjdV (11.19)
V
where BLj is a member of the matrix, B, of derivatives of the shape functions, N. If equilibrium is maintained from the discretised point of view, these internal forces are equal to external forces (if external forces are distributed, a weighting is necessary):
Fun = FLX (11.20)
Similarly, for diffusion phenomena the nodal internal fluxes are equivalent to the local fluxes:
Fnt = f [SNl — f diNu]dV (11.21)
V
If the balance equation is respected from the discretised point of view, these internal fluxes are equal to external ones:
FLnt = F[xt (11.22)
However, as we are considering non linear-problems, equilibrium/balance cannot be obtained immediately, but requires iteration. This means that the equations (Eqs. 11.20 and 11.22) are not fulfilled until the last iteration of each step.
Non-linear problems have been solved for some decades, and different methods have been used. From the present point of view, the Newton-Raphson method is the
reference method and probably the best one for a large number of problems. Let us describe the method. In Eq. 11.20 the internal forces Fg/ are dependant on the basic unknown of the problem, i. e. the displacement field. Similarly in Eq. 11.22 the internal fluxes are dependant on the pressure (temperature, concentration…) field.
If the external forces/fluxes don’t equilibrate, the question to be treated can be formulated in the following manner. Following the Newton-Raphson method, one develops the internal force as a first order Taylor’s series around the last approximation of the displacement field:
— Fint
Fu — FL (h)) + + O2 = FLx, (11.23)
-/Kj
where the subscript (i) indicates the iteration number and O2 represents second order, infinitely small terms. This is a linearization of the non-linear equilibrium equation. It allows one to obtain a correction of the displacement field:
/А Fint 1
-g FLfQa)) — FUX) — ELikj {F^Qi)) — FUT) (11.24)
Here, the matrix, E, represented here by its member term ELi, Kj, is the so-called stiffness matrix. With the corrected displacement field, one may evaluate new strain rates, new stress rates, and new improved internal forces. Equilibrium should then be improved.
The same meaning may be developed for diffusion problems using Taylor’s development of the internal fluxes with respect to the pressure/temperature/concen — tration nodal unknowns.
The iterative process may be summarised as shown in Fig. 11.1 for a one-d. o.f. solid mechanics problem. Starting from a first approximation of the displacement field /(i) the internal forces Fint (1) (point A(1) in the figure) are computed to be lower then the imposed external forces Fext. Equilibrium is then not achieved and a new approximation of the displacement field is sought. The tangent stiffness matrix is
evaluated and an improved displacement is obtained l(2) (point B(i)) (the target being as in Eq. 11.22. One computes again the internal forces Fint(2) (point A(2)) that are again lower then the external forces Fext. As equilibrium is not yet fulfilled, a new approximation of the displacement field is sought, l(3) (point B(2)). The procedure has to be repeated until the equilibrium/balance equation is fulfilled with a given accuracy (numerical convergence norm). The process has a quadratic convergence, which is generally considered as the optimum numerical solution.
However the Newton-Raphson method has an important drawback: it needs a large amount of work to be performed as well as to be run on a computer. The stiffness matrix, E, is especially time-consuming for analytical development and for numerical inversion. Therefore other methods have been proposed:
• An approximate stiffness matrix, in which some non-linear terms are neglected.
• Successive use of the same stiffness matrix avoiding new computation and inversion at each iteration.
It should be noted that each alternative is reducing the numerical convergence rate. For some highly non-linear problems, the convergence may be lost, and then no numerical solution will be obtained.
Some other authors, considering the properties and the efficiency of explicit time schemes for rapid dynamic problems (e. g. for shock modelling) add an artificial mass to the problem in order to solve it as a quick dynamic one. It should be clear that such a technique might degrade the accuracy of the solution, as artificial inertial effects are added and the static equilibrium Eq. 11.1 is not checked.
