1. i i row1 row2 row3 row4 col1 col2 col3 col4 a. Grid b. Mesh c. Cloud A control volume solution based on an unstructured mesh (Linear Triangular Elements) Control Volume Finite Element CVFE Method With boundary conditions We will use our standard steady state test problem First let us outline the basic ideas Can readily handle arbitrary domains

2. i i i,1 i,2 i,3 i,4 i,5 i,6 i,7 Region of SupportControl Volume i First Geometric features Or in more detail

3. i control volume support j = 4 k = S i,4 j = 1 or 5 k = S i,1 or S i,5 j = 2 k = S i,2 j = 3 k = S i,3 i internal node j = 1 k = S i,1 N i = 4 boundary node j = 2 k = S i,2 j = 3 k = S i,3 S i,,5 = 0 j = 4 k = S i,4 N i = 4 element f1 nf1nf1 f2

4. i control volume support j = 4 k = S i,4 j = 1 or 5 k = S i,1 or S i,5 j = 2 k = S i,2 j = 3 k = S i,3 internal node N i s = 4 For control volume shown our governing equation can be written as Length of face Calculate this From node value In element element n The element shown will contribute to the coefficients

5. 3 The task calculate this quantity In terms of nodal values of T At the mid point Of each of the faces Of the control volume Two bits of information needed 1. The linear interpolation in the element via the shape function Note we can approximate the continuous variable in the element as So in element unknown function is a planar surface. Note a, b and c are constants chosen such that values of T are piecewise (C 0 ) continuous between elements --“reptile skin” approximation 2 See Shape Function notes---from web—shape.pdf