Dear colleagues,
I am curious about the criterion employed by ABAQUS regarding the interpolation of the nodal temperature vaues within each finite element. As most of you already know, the temperature field specified by the user gives temperature values to the nodes and afterwards ABAQUS interpolates these temperature values to the Gauss integration points.
As the manual says, "the shape functions for interpolation of the temperature field may be different from the interpolation functions for the displacements; for example, if the underlying elements are of second order, the displacements are interpolated using quadratic functions, whereas the temperature field is interpolated using linear shape functions", this is consistent since (again quoting the manual) "the temperature interpolation in the stress elements is usually approximate and one order lower than the displacement interpolation to obtain a compatible variation of thermal and mechanical strain".
However, when using quadratic elements, temperature values at the integration points are not obtained directly from a linear interpolation of adjacent nodal values but from the sum of all the nodal temperature values in the element multiplied by some weights.
For example, when using a single CPS8 element and asigning a temperature of 1 to the node 1 (numbering according to the manual) and 0 to the other nodes, we obtain the following temperature distribution in the integration points:
-0.2333 (7) -0.1479 (8) -0.0624 (9)
-0.0188 (4) -0.0833 (5) -0.1479 (6)
0.1958 (1) -0.0188 (2) -0.2333 (3)
This is due to the weights I've mentioned before, and therefore, the temperature of the 1st integration point is obtained based on the following expression:
T*(1) = T(1) x 0,1958 + T(2) x (-0,2333) + T(3) x (-0,0624) + T(4) x (-0,2333) + T(5) x 0,5915 + T(6) x 0,0751 + T(7) x 0,0751 + T(8) x 0,5915
Where T(X) is the corresponding nodal temperature value. Therefore, my question is, how is the value of these weights obtained? What grounds are behind this choice? Is there any phisical meaning?
Thanks a lot for your help, I really appreciate it.
I am curious about the criterion employed by ABAQUS regarding the interpolation of the nodal temperature vaues within each finite element. As most of you already know, the temperature field specified by the user gives temperature values to the nodes and afterwards ABAQUS interpolates these temperature values to the Gauss integration points.
As the manual says, "the shape functions for interpolation of the temperature field may be different from the interpolation functions for the displacements; for example, if the underlying elements are of second order, the displacements are interpolated using quadratic functions, whereas the temperature field is interpolated using linear shape functions", this is consistent since (again quoting the manual) "the temperature interpolation in the stress elements is usually approximate and one order lower than the displacement interpolation to obtain a compatible variation of thermal and mechanical strain".
However, when using quadratic elements, temperature values at the integration points are not obtained directly from a linear interpolation of adjacent nodal values but from the sum of all the nodal temperature values in the element multiplied by some weights.
For example, when using a single CPS8 element and asigning a temperature of 1 to the node 1 (numbering according to the manual) and 0 to the other nodes, we obtain the following temperature distribution in the integration points:
-0.2333 (7) -0.1479 (8) -0.0624 (9)
-0.0188 (4) -0.0833 (5) -0.1479 (6)
0.1958 (1) -0.0188 (2) -0.2333 (3)
This is due to the weights I've mentioned before, and therefore, the temperature of the 1st integration point is obtained based on the following expression:
T*(1) = T(1) x 0,1958 + T(2) x (-0,2333) + T(3) x (-0,0624) + T(4) x (-0,2333) + T(5) x 0,5915 + T(6) x 0,0751 + T(7) x 0,0751 + T(8) x 0,5915
Where T(X) is the corresponding nodal temperature value. Therefore, my question is, how is the value of these weights obtained? What grounds are behind this choice? Is there any phisical meaning?
Thanks a lot for your help, I really appreciate it.