Numerical Analysis Questions Medium
The shooting method is a numerical technique used to solve boundary value problems (BVPs) in the context of the finite element method (FEM). BVPs involve finding the solution to a differential equation subject to specified boundary conditions.
In the shooting method, the BVP is transformed into an initial value problem (IVP) by introducing an artificial parameter. This parameter is used to control the boundary conditions and is adjusted iteratively until the desired boundary conditions are satisfied.
The steps involved in the shooting method for solving BVPs with FEM are as follows:
1. Discretization: The domain of the problem is divided into a finite number of elements, and the solution is approximated by a piecewise continuous function within each element. This is done by selecting appropriate basis functions and interpolating the solution within each element.
2. Formulation: The differential equation is transformed into a system of algebraic equations using the finite element method. This involves multiplying the differential equation by a weight function and integrating over each element. The resulting equations are then assembled into a global system of equations.
3. Initial Guess: An initial guess for the artificial parameter is chosen, which corresponds to an initial guess for the solution. This guess should satisfy the boundary conditions.
4. Integration: The IVP is solved by integrating the system of algebraic equations in time, starting from the initial guess. This is typically done using numerical integration methods such as the Euler method or the Runge-Kutta method.
5. Residual Evaluation: The solution obtained from the integration is evaluated at the final time step, and the residual is computed. The residual represents the deviation of the solution from the desired boundary conditions.
6. Parameter Adjustment: The artificial parameter is adjusted based on the residual, using an iterative method such as the Newton-Raphson method. The goal is to minimize the residual and bring the solution closer to the desired boundary conditions.
7. Convergence Check: Steps 4-6 are repeated until the residual falls below a specified tolerance level, indicating convergence. At this point, the solution is considered to have satisfied the boundary conditions.
8. Post-processing: Once the solution has converged, it can be further analyzed and post-processed to extract any desired quantities of interest, such as stresses, strains, or other physical quantities.
Overall, the shooting method in combination with the finite element method provides a powerful numerical approach for solving boundary value problems, allowing for accurate and efficient solutions to a wide range of engineering and scientific problems.