Nonlinear finite elements/Updated Lagrangian approach

From Wikiversity
Jump to navigation Jump to search

Updated Lagrangian Approach[edit]

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 Measures[edit]

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 Form[edit]

The Updated Lagrangian governing equations are:

Conservation of Mass[edit]

For the rod,

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

Conservation of Momentum[edit]

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

Conservation of Energy[edit]

If we ignore heat flux and heat sources

If we include heat flux and heat sources

In short form:

Constitutive Equations[edit]

Total Form[edit]

For a linear elastic material:

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

For small strains:

Rate Form[edit]

For a linear elastic material:

Initial and Boundary Conditions for Updated Lagrangian[edit]

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 Lagrangian[edit]

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 Lagrangian[edit]

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 Term[edit]

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 Term[edit]

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 Terms[edit]

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 Lagrangian[edit]

The finite element equations in updated Lagrangian form are: