This brief write up attempts to explain the physical interpretation of carrying out integration by parts during the formualtion of finite element (FE) equations using Galerkin's weighted residual approach.
During the formulation of FE equations using the Galerkin's method of weighted residual, we carry out the following steps;
- Obtain the governing differential equation
- Express the approximate solutuon using trial functions
- Obtain the residue by substituting the approximate solution in the governing differential equation
- Express the Galerkin's weighted residual statement i.e. the inner product of the weighting functions and the residue when integrated over the whole domain is zero.
- Integrate the above mathematical expression by parts.
The integration by parts is carried out repeatedly as long as the boundary terms are physically meaningful. In structural mechanics problems the boundary terms represent the work done by end forces in moving through end displacements. Thus for truss/rod problems the integration by parts is performed just once whilst it is performed twice for Euler-Bernoulli beam problems.
An off-shoot of this requirement for structural mechanics problems is that the order derivative on the weighting function term is same as the order of derivative on the primary variable --- leading to what is known as "symmetric bilinear form" --- which in turn is responsible for symmetric stiffness matrix.
Without carrying out the integration by parts the gradient boundary condition (which correspond to end force in truss/bar problems or end shear in beam problems) could not have been obtained in the FE equations and the symmetricity of the stiffness matrix would not result.
Thus it is the step of "integration by parts" that results in the terms which comprise the force vector of the equilibrium equation {F} = [k] {u} and results in a symmetric stiffness matrix.
