Text Book on Dynamic Relaxation Method

(osama mohammed elmardi suleiman khayal) #1

The method of dynamic relaxation in its early stages of development was
perceived as a numerical finite difference technique. It was first used to analyze
structures, then skeletal and cable structures, and plates. The method relies on a
discretized continuum in which the mass of the structure is assumed to be
concentrated at given points (i.e. nodes) on the surface. The system of concentrated
masses oscillates about the equilibrium position under the influence of out of balance
forces. With time, it comes to rest under the influence of damping. The iterative
scheme reflects a process, in which static equilibrium of the system is achieved by
simulating a pseudo dynamic process in time. In its original form, the method makes
use of inertia term, damping term and time increment.
Chapter one includes introduction to dynamic relaxation method (DR) which is
combined with finite differences method (FD) for the sake of solving ordinary and
partial differential equations, as a single equation or as a group of differential
equations. In this chapter the dynamic relaxation equations are transformed to
artificial dynamic space by adding damping and inertia effects. These are then
expressed in finite difference form and the solution is obtained through iterations.
In chapter two the procedural steps in solving differential equations using DR
method were applied to the system of differential equations (i.e. ordinary and/ or
partial differential equations). The DR program performs the following operations:
Reads data file; computes fictitious densities; computes velocities and displacements;
checks stability of numerical computations; checks convergence of solution; and
checks wrong convergence. At the end of this chapter the dynamic relaxation (DR)
numerical method coupled with the finite differences discretization technique is used
to solve nonlinear ordinary and partial differential equations. Subsequently, a
FORTRAN program is developed to generate the numerical results as analytical and/
or exact solutions.
Chapter three discusses the importance of using numerical methods in solving
differential equations and stresses the use of dynamic relaxation technique in these
The book is suitable as a textbook for a first course on dynamic relaxation
technique in civil and mechanical engineering curricula. It can be used as a reference
by engineers and scientists working in industry and academic institutions.
Assistant Professor
Osama Mohammed Elmardi SuleimanText Book on Dynamic Relaxation Method.pdf (1.0% u)

(عمارة إسلامية) #2

بسم الله الرّحمن الرّحيم
السّلام عليكم ورحمة الله وبركاته

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