Nonlinear finite elements/Updated Lagrangian approach

Updated Lagrangian ApproachEdit

For the total Lagrangian approach, the discrete equations are formulated with respect to the reference configuration. For the updated Lagrangian approach, the discrete equations are formulated in the current configuration, which is assumed to be the new reference configuration.

The independent variables in the total Lagrangian approach are   and  . In the updated Lagrangian they are   and   which are with respect to the new reference configuration.

The dependent variable in the total Lagrangian approach is the displacement  . In the updated Lagrangian approach, the dependent variables are the Cauchy stress   and the velocity  .

Updated Lagrangian Stress and Strain MeasuresEdit

The stress measure is the Cauchy stress:

 

The strain measure is the rate of deformation (also called velocity strain):

 

Note that the derivative is a spatial derivative. It can be shown that

 

where   is the deformation gradient. So the integral of the rate of deformation in 1-D is similar to the logarithmic strain (also called natural strain).

Governing Equations in Updated Lagrangian FormEdit

The Updated Lagrangian governing equations are:

Conservation of MassEdit

 

For the rod,

 

These are the same as those for the total Lagrangian approach.

Conservation of MomentumEdit

 

In this case   may not be constant at along the length and further simplification cannot be done. In short form,

 

Conservation of EnergyEdit

If we ignore heat flux and heat sources

 

If we include heat flux and heat sources

 

In short form:

 

Constitutive EquationsEdit

Total FormEdit

For a linear elastic material:

 

The superscript   refers to the fact that this function relates   and  .

For small strains:

 
Rate FormEdit

For a linear elastic material:

 

Initial and Boundary Conditions for Updated LagrangianEdit

For the updated Lagrangian approach, initial conditions are needed for the stress and the velocity. The initial displacement is assumed to be zero.

The initial conditions are:

 

The essential boundary conditions are

 

The traction boundary conditions are

 

The unit normal to the boundary in the current configuration is  . The tractions in the current configuration are related to those in the reference configuration by

 

The interior continuity or jump condition is

 

Weak Form for Updated LagrangianEdit

The momentum equation is

 

To get the weak form over an element, we multiply the equation by a weighting function and integrate over the current length of the element (from   to  ).

 

Integrate the first term by parts to get

 

Plug the above into the weak form and rearrange to get

 

If we think of the weighting function   as a variation of   that satisfies the kinematic admissibility conditions, we get the principle of virtual power:

 

Recall that

 

Therefore,

 

and

 

Hence we can alternatively write the weak form as

 

Comparing with the energy equation, we see that the first term above is the internal virtual power. Using the substitutions   and  , we can also write the weak form as

 

The weak form may also be written in terms of the virtual powers as

 

where,

 

Finite Element Discretization for Updated LagrangianEdit

The trial velocity field is

 

In matrix form,

 

The resulting acceleration field is the material time derivative of the velocity

 

Hence,

 

Note that the shape functions are functions of   and not of  . We could transform them into function of   using the inverse mapping

 

However, in that case the shape functions become functions of time and the procedure becomes more complicated.

The test (weighting) function is

 

The derivatives of the test functions with respect to   are

 

It is convenient to use the isoparametric concept to compute the spatial derivatives. Let us reexamine what this approach involves.

Figure 1 shows a two-noded 1-D element in the reference and current configurations along with the parent element.

 
Figure 1. Reference and Current Configurations of a 1-D element.

The map between the Eulerian coordinates and the parent coordinates is

 

At  ,

 

Therefore the displacement in the parent coordinates is

 

Similarly, the velocity and its variation in the parent coordinates are given by

 

The acceleration in the parent coordinates is given by

 

The rate of deformation is given by

 

We can evaluate the derivative with respect to   by simply using the map to the parent coordinate instead of mapping back to the reference coordinates. Using the chain rule (for the two noded element)

 

In matrix form,

 

The rate of deformation in parent coordinates is then given by

 

To derive the finite element system of equations, we follow the usual approach of substituting the trial and test functions into the approximate weak form

 

Let us proceed term by term.


First LHS TermEdit

The first term represents the virtual internal power

 

Plugging in the derivative of the test function, we get

 

The matrix form of the expression for virtual internal work is

 

The internal force is

 

Note that the above simplification only occurs in 1-D. In matrix form,

 

Remark:

Note that we can write

 

If we use the above relation, we get

 

Using the relation

 

we get

 

The above is the same as the expression we had for the internal force in the total Lagrangian formulation.

Second LHS TermEdit

The second term represents the virtual kinetic power

 

Plugging in the test function, we get

 

The inertial (kinetic) force is

 

Now, plugging in the trial function into the expression for the inertial force, we get

 

where

 

and

 

In matrix form,

 

The consistent mass matrix in matrix notation is

 

Plugging the expression for the inertial force into the expression for virtual kinetic power, we get

 

In matrix form,

 

Remark:

Note that since the integration domain is a function of time, the mass matrix for the updated Lagrangian formulation is also a function of time. However, from the conservation of mass, we have

 

Therefore, we can write the mass matrix as

 

Hence the mass matrices are the same for both formulations.


RHS TermsEdit

The right hand side terms represent the virtual external power

 

Plugging in the test function into the above expression gives

 

In matrix notation,

 

The external force is given by

 

In matrix notation,

 

Remark:

Using the conservation of mass

 

and the relations between the traction sin the reference and the current configurations

 

we can transform the integral in the expression for the external force to one over the reference coordinates as follows:

 

The above is the same as the expression we had for the external force in the total Lagrangian formulation.

Discrete Equations for Updated LagrangianEdit

The finite element equations in updated Lagrangian form are: