C1. Mechanical non-linear problems

Description of possible non-linearities

The non-linearity of a mechanical problem comes from the fact that the coefficients of the equilibrium equation depend on the displacements of the solid itself in equilibrium. In other words, the equilibrium equation is generally implicit.

There are several non-linearity categories for mechanical static problems:

Material non-linearities: in cases where the constitutive law is not linear or that the response of the material depends on the loading history. In other words, stress is not a linear function of strains. The most common case in civil engineering is one of the material loaded beyond their elastic capacity, which then develops elasto-plastic behaviors. This behavior is characterized by a dependency between the stiffness of the material and its stress state.

Geometric nonlinearities: for cases where the structure is submitted to big displacements or big strains. In the first case, one can no longer write the problem neglecting the changes in the geometry of the structure. In the second case, one can no longer approach the strains simply as the gradient of the displacements.

Boundary condition non-linearities: in cases where a structure is progressively loaded and there is potential contact between two bodies with follower forces. These types of non-linearities also appear when the construction phasing or the assembly of a bridge’s deck are simulated, when digging a gallery, constructing an embankment, etc.

All the above non-linearities can be coupled if the algorithm allows it, but the resolution of the problem becomes more complex.

Principle of resolution of a non-linear problem: Newton method

When solving a finite element problem, one looks for the displacement field u, such that the inner forces L_int are equal to the external forces L_ext:

, which is a non-linear problem as a function of u.

Generally, to solve the non-linear static problem, an incremental algorithm is used. For that matter, the problem is parameterized in terms of t (with t representing a pseudo-time, unlike the t parameter used in dynamics). This parameter is used to index the successive load-steps applied on the structure. More precisely, it consists of searching for the equilibrium state corresponding to the successive load-steps F₁, F₂, …

This separation leads to solving a series of quasi-linear problems as shown in the figure below and to determine the state of the structure at the time-step t (displacements, strains, stresses) knowing the solution at the state t-1. The greater the number of load-steps, the better the precision.

time

increment.

At each increment t_i the discrete problem is K_i x q_i = F_i where q_i is the unknown displacement vector under the applied imposed loading F_i. While in the linear case seen in chapter 1 the K matrix was explicit, when the problem is non-linear, K_i is a matrix with its terms depending implicitly on the value of q_i. So, q_i cannot be determined directly by computing the inverse of the matrix K.

The most used method to solve this non-linear equation is to use a Newton-type algorithm. The idea is to build a good approximation for the equation’s solution

considering its first-order Taylor expansion

One must start from an initial point (close enough to the solution) and then compute by iterations

At each iteration, one should evaluate the residual vector F(qk) until it exceeds (in absolute value) the value arbitrarily close to zero. This convergence criterion must be chosen with care, respecting the standard used by the calculation code (see section 3.3 for more details).

Note: With the Newton method, at each iteration, one should compute the tangent matrix at the considered point:

The computational cost of this matrix can be time-consuming. If using this matrix allows having a quadratic convergence (so, in fewer iterations), it is not essential to use this matrix. Other strategies can be adopted to estimate this matrix, namely the quasi-Newton methods. It is conceivable to use the tangent matrix without updating it at each iteration, but also to use the elastic matrix (figure b) or the secant matrix in the case of a damage model. An illustration of the successive iterations according to the used matrix is shown below.

Illustration of the Newton or quasi-Newton method (elastic matrix)

In general, using the tangent matrix allows a faster convergence (in fewer iterations) but the alternatives might be more effective or more robust according to the situation.

As the method is iterative, the process should be stopped when the stop criterion is reached, in other words, when it is verified that a given value (or several values) becomes negligible. The global algorithm can be written as follows:

by defining the increments, i indexing the Newton iterations and ε being a positive value close to zero.

Note: The Newton algorithm is used to solve the equilibrium at each time step. It can also be used to find the stresses in each Gauss point (at all iterations of the Newton problem on the global scale) when the constitutive law requires it.

by defining the increments, i indexing the Newton iterations and ε being a positive value close to zero.

Note: The Newton algorithm is used to solve the equilibrium at each time step. It can also be used to find the stresses in each Gauss point (at all iterations of the Newton problem on the global scale) when the constitutive law requires it.