You possibly mean the difference between the Linear or Quadratic shape functions for the approximations, because in 2D shape of the finite element can be both triangular and quadrilateral for various types of approximations (linear, quadratic, etc. high-order)

The main topics for answers can be summarized as follows:

1) The high order approximation for the finite element (keeping the same size) leads to the small error for the solution if all parameters (boundary conditions, geometry, materials) are sufficiently smooth. Thus the quadratic approximation is better than linear one.

--> Effect of p-refinement.

2) Triangular shapes for for FE of low order (linear) leads to the larger error (locking for bending)

--> Effect of the "mapped" "quadrilateral" mesh

3) Small size of FE leads to the smaller error (but it leads to many FE's)

--> Effect of h-refinement.

3) If the parameters are not smooth (singularities in geometry such as edges, or in boundary conditions) than combinations of the low order FE around these singularities together with high order FE somewhat far from singularities leads to the both optimal small error and number of DOF

--> Effect of hp-refinement

You possibly mean the difference between the Linear or Quadratic shape functions for the approximations, because in 2D shape of the finite element can be both triangular and quadrilateral for various types of approximations (linear, quadratic, etc. high-order)

The main topics for answers can be summarized as follows:

1) The high order approximation for the finite element (keeping the same size) leads to the small error for the solution if all parameters (boundary conditions, geometry, materials) are sufficiently smooth. Thus the quadratic approximation is better than linear one.

--> Effect of p-refinement.

2) Triangular shapes for for FE of low order (linear) leads to the larger error (locking for bending)

--> Effect of the "mapped" "quadrilateral" mesh

3) Small size of FE leads to the smaller error (but it leads to many FE's)

--> Effect of h-refinement.

3) If the parameters are not smooth (singularities in geometry such as edges, or in boundary conditions) than combinations of the low order FE around these singularities together with high order FE somewhat far from singularities leads to the both optimal small error and number of DOF

--> Effect of hp-refinement

It is an extraordinary response, Alexander.

In addition to this... "Usually", the error with complex volumes is acceptable with 10-node tetrahedral elements.

Always the decision is dependent on the geometry and always with volumes use quadratic elements... Right Alexander?

Thank you Dr. Alexander. It was very helpful.

Depending on your needs, you may require a converged mesh. That is a mesh refined enough so that your numerical results do not vary (regarding a specified tolerance) when you decrease the mesh size.

If you increase the order of your elements you end up with more DOF on the same base mesh as intermediate nodes have to be added in the elements to be able to interpolate richer shape functions. Convergence is however quicker regarding the mesh size (in general) as each element can represent richer fields.

Be aware that the stresses evolve with an order lower than the element order, so that if you use linear elements, you will get constant stresses per element. This is responsible for locking problems.

Nowadays, it is usually recognized that mesh size versus accuracy optimum can be obtained with quadratic elements for classical problems.

I you have discontinuities in your mesh response, convergence can be peculiar whatever the order, and special enhancements can be used through X-FEM for example.

This is really not an easy question because it depends on may things.

Here are some guidelines:

To avoid shear locking, do not use linear triangular/tetrahedral elements in bending problems. If you use linear rectangular/hexahedral elements in bending problems, shear locking may also be severe. You can use these elements with reduced integration, but then look out for hourglassing. Incompatible mode elements may help, but only if they are regular.

In other words: In elastic (or general elliptic) problems (especially bending problems), it is usually better to choose quadratic shape functions. (In addition, there is also volumetric locking, which may be a problem if you model incompressible materials or sometimes in palsticity. To avoid this, you may use hybrid elements.)

In problems with contact, quadratic shape functions may be problematic in establishing the contact between an isoparametric element with a curved outline and a straight contact surface. In this case, linear elements might be better.

In problems where deformation strongly localised and is discontinuous (e.g., shear bands), linear elements might be better to capture this.

i have a question. i have a beam element with only displacemenet DOF and use Shape function quadratic order and i obtain the shape function for per direction now i want use this shape function for drive the stiffness matrix but i don't know how use the shape function  for make B in this formula (B' E B )

For general problems, use the mesh that best balances the accuracy of the phenomenon under study and the computational cost to solve the problem