FEM

Question 1

what is a neumann boundary condition? Explain bith types.

Answer 1

homogeneous Neumann boundary conditions, which is another way of stating zero traction, {t},that is
{t} = [σ]{n} = {0},
where [σ] is the Cauchy stress and {n} is the normal to the boundary. Homogeneous Neumann boundary conditions are also referred to as natural boundary conditions, because in the case of zero surface traction we do not have to do anything to impose the constraint - it is the natural condition if we don’t specify a boundary condition.
* inhomogeneous Neumann boundary conditions, which is where a non-zero traction is applied to the boundary, such as a distributed pressure as we looked at in Lecture 14. These conditions are imposed by integrating the tractions along the surface of an element to determine the equivalent nodal forces via
{f} = Integral ([N]T {t})dS

Question 2

explain dirichlet boundary conditions

Answer 2

Dirichlet, also known as displacement, boundary conditions or fixities. Dirichlet conditions12 enforce a
constraint on the primary unknown in a finite element simulation. For this course the primary unknown is
the displacement at the nodes, however if we were to conduct a thermal analysis Dirichlet constraints
would be imposed on the temperature at the nodes. In the literature you may also read about:
* homogeneous Dirichlet boundary conditions, which is another way of stating zero displacement
along the boundary.
* inhomogeneous Dirichlet boundary conditions, which is where a non-zero displacement is applied
to the boundary.
These displacement boundary conditions can be applied in a number of different ways, a commonly
adopted method is to impose the conditions strongly - see Lecture 13. They can also be imposed
weakly via a penalty-type approach, however these methods are beyond the scope of this course.

Question 3

what are the requirements for completness in an fem porblem

Answer 3

There are two requirements of completeness:
* the finite element approximation of the displacement vector, {u}, must be able to represent an arbitrary constant rigid-body motion; and
* the finite element approximation of the displacement vector, {u}, must be able to represent an arbitrary constant strain state

Question 4

what are the requirements for compatibility in an FEM problem

Answer 4

The compatibility (or conforming) condition requires that:
* the finite element approximation of the displacement vector, {u}, must vary in a continuous
manner over element boundaries.
All of these conditions place requirements on the finite element shape functions, [N], as they control
the displacement variation through an element through
{u} = [N]{d},

Question 5

what happens if both completeness and compatibility are met

Answer 5

if conditions are satisfied then we should obtain monotonic convergence towards the exact
solution for a problem as we refine the mesh.