Why Abaqus Fails to Find a Converged Solution in Non linear Problems

The basic problems….

• Many linear solutions are needed to minimize the residual

• We can’t predict beforehand how many linear solutions will be required

• Nonconvergence is possible

Meshing Methods in FEA

There are two main meshing methods in FEA: Free mesh and Mappped mesh

 

Free Mesh/Automated Mesh/Unstructured Mesh

•Has no element shape restrictions.

•The mesh does not follow any pattern.

•Suitable for complex shaped areas and volumes.

 

Free mesh

 

 

 

Mapped Mesh/Manual Mesh/Structured Mesh

•Restricts element shapes to quadrilaterals for areas and hexahedral (bricks) for volumes.

•Typically has a regular pattern with obvious rows of elements.

•Suitable only for “regular” areas and volumes such as rectangles and bricks.

 

Mapped mesh

How the results are achieved in FEA

•Formulation of the components 'stiffness' matrix

•This may be used to evaluate the displacements of the structure

•On evaluation of the displacements, they are differentiated to give six strain distributions, 3 mutually perpendicular direct strains & 3 corresponding shear strains.

•Finally six stress distributions are determined via the stress/strain relationships of the material.

•A point to note is that at least one of the displacements must be known before the rest can be determined (before the system of equations can be solved). These known displacements are referred to as boundary conditions

Basic steps in the FEA

Pre-processing phase

1. Geometric description of the model and discretization into nodes and elements.

2. Assume a shape function to represent the physical behaviour of an element.

3. Develop equations for an element.

4. Assemble the elements to present the entire problem. Construct the global stiffness matrix

5. Apply boundary conditions, initial conditions and loading.

 

Solution Phase

6. Solve a set of linear or non-linear algebraic equations simultaneously to obtain nodal results, such as displacement values at different nodes.

  [k]{q} = {F}

Post-processing phase

7. Obtain other important information like principal stresses, heat fluxes, etc…

8. Graphical visualization of results

FEA Solver background of Modal analysis

The equilibrium equation for a structure performing free vibration appears as the eigenvalue problem:

     [K-λM]x = 0

where K is the stiffness matrix of the structure and M is the mass matrix.  Damping is neglected.  The solution of the eigenvalue problem yields n eigenvalues λ, where n is the number of degrees of freedom.  The vector x is the eigenvector corresponding to the eigenvalue.

The eigenvalue problem is solved using a matrix method called the Lanczos method.  Not all eigenvalues are required -- only a small number of the lowest eigenvalues are normally calculated.

The natural frequency f_i follows directly from the eigenvalue λ.

 

In NASTRAN solver, in order to run a normal modes analysis, an EIGRL bulk data entry needs to be given because it defines the number of modes to be extracted.  The EIGRL card needs to be referenced by a METHOD statement in a SUBCASE in the subcase information section.

FEA Solver background of Linear static analysis

The basic finite element equation to be solved for structures experiencing static loads can be expressed as:

Ku = P

where K is the stiffness matrix of the structure (an assemblage of individual element stiffness matrices).  The vector u is the displacement vector, and P is the vector of loads applied to the structure.  The above equation is the equilibrium of external and internal forces.

The stiffness matrix is singular, unless displacement boundary conditions are applied to fix the rigid body degrees of freedom of the model.

The equilibrium equation is solved either by a direct or an iterative solver.  By default, the direct solver is invoked, whereby the unknown displacements are simultaneously solved using a Gauss elimination method that exploits the sparseness and symmetry of the stiffness matrix, K, for computational efficiency. Alternatively, an iterative solver using the preconditioning conjugate gradient method may be used.  While the direct solver is very robust, accurate and efficient, the iterative solver is sometimes superior, in terms of speed, for thick-walled solid structures. 

Once the unknown displacements at the nodal points of the elements are calculated, the stresses can be calculated by using the constitutive relations for the material.  For linear static analysis where the deformations are in the elastic range, i.e.: the stresses, σ, are assumed to be linear functions of the strains, ε, Hooke’s law can be used to calculate the stresses.  Hooke’s law can be stated as:

σ = Cε

with the elasticity matrix C of the material.  The strains ε are a function of the displacements.