Evaluating Element Mesh Conformance Criteria

[Larsson]

Homework Statement

An element mesh is based on the following 3-node and 4-node elements (0,0) (0.5,0.5) (0,1) (1.5,0) (1.5,0.5). The approximation for the 4-node elements is T = a1 +a2x +a3y +a4xy, while the approximation for the 3-node element is T = b1 +b2x +b3y. Is the converegence criterion fulfilled?

The Attempt at a Solution

I looked in the book and found that for the convergence criterion both completeness and compatibility or conforming requirement be satisfied.

Completeness:
*the approximation of the displacement vector u must be able to represent an arbitrary constant rigid-body motion

*the approximation of the displacement vector u must be able to represent an arbitrary constant strain state

compatibility or conforming requirement.:
*The approximation of the displacement vector u must vary in a continuous manne over element boundaries.


The solution says
The completeness requirement is fulfilled for both elements. To fulfill the compatibility requirement the approximated field must be continues i.e the approximation must be uniquely determined by the nodal values on the boundaries. This is not satisfied for the current configuration, i.e compatibility is not satisfied.

So, here I have both the question and the solution, the problem is that I don't understand it. Can anyone help? I don't really understand how to check for the criterias and what they really mean.

 

[mighty2000]

Hello,

Thank you for your question. I am happy to help you understand the convergence criterion for element meshes.

The convergence criterion is an important aspect in finite element analysis, which ensures that the solutions obtained are accurate and reliable. It consists of two main requirements: completeness and compatibility.

Completeness refers to the ability of the approximation to represent an arbitrary constant rigid-body motion and an arbitrary constant strain state. In simpler terms, it means that the approximation should be able to capture any type of deformation or motion that may occur in the system being analyzed.

Compatibility, on the other hand, refers to the continuity of the approximated field over element boundaries. This means that the approximation should be uniquely determined by the nodal values on the boundaries. In other words, the values of the approximation at the nodes should match and form a continuous field over the element boundaries.

In the given problem, the completeness requirement is fulfilled for both the 3-node and 4-node elements. This means that the approximations for both elements are able to represent any type of deformation or motion. However, the compatibility requirement is not fulfilled for the current configuration. This means that the approximated field is not continuous over the element boundaries and therefore, the approximation is not uniquely determined by the nodal values on the boundaries.

In conclusion, the convergence criterion is not fulfilled for the given element mesh. This may lead to inaccurate results and the need for further refinement of the mesh.

I hope this explanation helps you understand the convergence criterion better. If you have any further questions, please feel free to ask. Keep up the good work in your studies!