Dynamic Relaxation Method. Theoretical Analysis, Solved Examples and Computer Programming

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Chapter 2.1.1 Numerical Instability:
In every iteration, the value of the function at the center of the solution domain or other suitable point is compared with two estimated reference values representing lower and upper bounds of the function at that point. If solution was failed such that the computed value of the function at the specified point did not fall within the prescribed range, the solution is deemed unstable, and therefore iterations are terminated. The damping coefficients are then reduced and the process of iteration is restarted once again. The iterations are repeated several times until stability is reached.
2.1.2 Convergence of DR Solution:
Convergence of the dynamic relaxation solution is checked at the end of each iteration by comparing the velocities over the domain with a prescribed value. The procedure is repeated until the solution is deemed converged and consequently the iterative process is terminated.
2.1.3 Convergence to an Invalid Solution:
Sometimes DR solution converges to incorrect answer. Check for invalid solution is carried out after the solution has satisfied the convergence criterion explained earlier. In the check procedure the profile of variable is compared with the anticipated profile over the domain. For instance, if the value of the function on the boundaries is zero, and it is known that the function increases from edge to center, then the solution should follows a similar profile. If the computed profile is different from that, the solution is deemed to be incorrect. When this happens, the solution can hardly be made to converge to the correct answer by altering the damping coefficients and time increment. One should take another look to the boundary conditions and correct them if they are wrong.
2.1.4 Time Increment:
Proper time increment is a very important factor for speeding convergence and controlling numerical computations. When time increment is too small, convergence becomes tediously slow; and

Osama Mohammed Elmardi was born in Sudan in 1966. He received a bachelor degree in mechanical engineering from SUST University Sudan in 1998 and a master degree in solid mechanics from Nile Valley University in 2003.