Difference strong form and weak form in finite element analysis
Answers
Answered by
0
A Strong Form of differential equation is one which usually deals with the original governing equation of the physical problem, with no mathematical manipulation as such. That being said, sometimes, such problems are difficult to handle and at times, may even be infeasible.I would illustrate this with an example.
When encountered with a Beam Problem loaded with multiple transverse loads, the solution by Galerkin’s Method becomes increasingly difficult, because of the following reasons -
The assumed solution must be a biquadratic polynomial (please remember that it can be of a higher order, but not lower !). This is because, the governing equation for a Beam problem is a fourth order differential equation. Thus, we need to have the values of five constants (namely, a0,a1,a2,a3,a4). You can very well guess the mathematical calculations involved.The residual function when multiplied with the weight functions (remember that Finite Element Analysis uses the shape functions which are equivalent to the weight functions usually), would give a very large expression, which needs to be integrated over the entire domain. Such integrals are impractical to be carried out on Standard Scientific Calculators (which by the way, would easily run out of memory !). So unless you are equipped with a high-end computational software like MATLAB or SciPy library of Python, you are in some real deep!
Thus, we need to have a mathematical manipulation which reduces the required degree of the polynomial function (assumed solution) and eases the calculations. ENTER, the Weak Form of the differential equation
When encountered with a Beam Problem loaded with multiple transverse loads, the solution by Galerkin’s Method becomes increasingly difficult, because of the following reasons -
The assumed solution must be a biquadratic polynomial (please remember that it can be of a higher order, but not lower !). This is because, the governing equation for a Beam problem is a fourth order differential equation. Thus, we need to have the values of five constants (namely, a0,a1,a2,a3,a4). You can very well guess the mathematical calculations involved.The residual function when multiplied with the weight functions (remember that Finite Element Analysis uses the shape functions which are equivalent to the weight functions usually), would give a very large expression, which needs to be integrated over the entire domain. Such integrals are impractical to be carried out on Standard Scientific Calculators (which by the way, would easily run out of memory !). So unless you are equipped with a high-end computational software like MATLAB or SciPy library of Python, you are in some real deep!
Thus, we need to have a mathematical manipulation which reduces the required degree of the polynomial function (assumed solution) and eases the calculations. ENTER, the Weak Form of the differential equation
Similar questions