ANSYS generate two type of solution: Nodal solution and Elemental solution. Is the results generated by nodal solution is though nodal integration and the results generated by elemental solution is though Gauss point integration. Or both the results are generated by nodal solution but in elemental solution results were averaged to nodal value. I care fully analyze both the solution and i found that there is difference between two values.
First the FEA computation process in ANSYS:
We have two type of nodes involved in an FEA solution. Example shown (please see the attachment) is a plate with circular hole and horizontal tensile load applied at the two ends. Due to symmetry only a quadrant is simulated using (say) ANSYS. We have shown triangular elements. There are two type of nodes:
Local nodes (shown inside the elements)
Global nodes (shown outside in blue color)
First stiffness matrices are formed at element level. For example the element stiffness matrices for the elements ‘a’ and ‘b’ can be written in terms of local load vectors [r1 r2 r3]aT, [r1 r2 r3]bT, local stiffness values kij, and local displacement vectors [d1 d2 d3]aT , [d1 d2 d3]bT. Here 1, 2, 3 denote the local nodes for the elements ‘a’ and ‘b’ .
Then by applying the fundamental concepts of Equilibrium and Compatibility we assemble the element stiffness matrices for all elements over the domain to obtain the global stiffness matrix system as shown in the figure attached. Here [D1 D2 …….DN]T and [R1 R2 …….RN]T are the global displacement and load vectors in terms of global nodes.
The overall problem is solved in terms of global nodes and solution is obtained as [D] = [K]-1[R]. Once displacement vector is computed we can compute strain components using the spatial derivatives of the displacement. Once strain at each node (global as well as local nodes) is known we can compute stress σ at each node.
ANSYS displays stresses in two ways (Please see the attachment):
Element stress solution and Nodal stress solution (see the attached figure)
Element stresses solutions: Here the element stresses are displayed within individual elements. They are non-averaged stresses. The stress distribution is unique to an individual element. Understanding local node distribution is important for this.
Nodal stress solutions: Here stresses are given to you in averaged form at each global node. The stress value at a global node is average of all the local-node-stress-values of all the elements sharing that global node as explained in the table in the attached figure.
The discontinuous type of stress distribution in element solution only indicates that you need mesh refinement. As you go on refining the mesh, the element solution looks more and more like the nodal solution.
Sir,
The displacements and rotation if present are calculated at each node but the stress and strains are evaluated at the element integration points. Then these results are extrapolated to the nodes. As you may know each node is associated with different element, so the nodes can have multiple values of stress and strains from each element. Thus, the average of those value is considered as nodal solution and they are continuous across the elements. Whereas the element solution have unaverage stress and strains values which are discontinuous across the elements. In simple nodal solutions are averaged value and element solutions are unaveraged values.
You can also see the contours of the nodal and element solution. The contours of the nodal solution are well and continuous between the elements but the elements solution contours will be disjoint or discontinuous. If you want both nodal and element solution to be more or less equal then take a finer mesh.
First the FEA computation process in ANSYS:
We have two type of nodes involved in an FEA solution. Example shown (please see the attachment) is a plate with circular hole and horizontal tensile load applied at the two ends. Due to symmetry only a quadrant is simulated using (say) ANSYS. We have shown triangular elements. There are two type of nodes:
Local nodes (shown inside the elements)
Global nodes (shown outside in blue color)
First stiffness matrices are formed at element level. For example the element stiffness matrices for the elements ‘a’ and ‘b’ can be written in terms of local load vectors [r1 r2 r3]aT, [r1 r2 r3]bT, local stiffness values kij, and local displacement vectors [d1 d2 d3]aT , [d1 d2 d3]bT. Here 1, 2, 3 denote the local nodes for the elements ‘a’ and ‘b’ .
Then by applying the fundamental concepts of Equilibrium and Compatibility we assemble the element stiffness matrices for all elements over the domain to obtain the global stiffness matrix system as shown in the figure attached. Here [D1 D2 …….DN]T and [R1 R2 …….RN]T are the global displacement and load vectors in terms of global nodes.
The overall problem is solved in terms of global nodes and solution is obtained as [D] = [K]-1[R]. Once displacement vector is computed we can compute strain components using the spatial derivatives of the displacement. Once strain at each node (global as well as local nodes) is known we can compute stress σ at each node.
ANSYS displays stresses in two ways (Please see the attachment):
Element stress solution and Nodal stress solution (see the attached figure)
Element stresses solutions: Here the element stresses are displayed within individual elements. They are non-averaged stresses. The stress distribution is unique to an individual element. Understanding local node distribution is important for this.
Nodal stress solutions: Here stresses are given to you in averaged form at each global node. The stress value at a global node is average of all the local-node-stress-values of all the elements sharing that global node as explained in the table in the attached figure.
The discontinuous type of stress distribution in element solution only indicates that you need mesh refinement. As you go on refining the mesh, the element solution looks more and more like the nodal solution.
Usually the numerical integration is performed via integration points "Quadrature" even the variables are stored on nodes, this is true especially for displacement vector. But in any case, transformations "interpolations" can always be made. The difference between nodes values and elements ones come from this interpolation. Moreover, the difference increases when the field gradient changes severely e.i. close to singularities ...
Regards
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