Copyright c  2005 Tech Science Press CMES, vol.8, no.3, pp.191-200, 2005 Elastodynamics with the Cell Method F. Cosmi 1 Abstract: The Cell Method is a recently developed nu- merical method that is giving interesting results in several fields of physics and engineering. In this paper, first a brief description of the method for elasticity problems is given and successively the elastodynamics formulation is derived. The method leads to an explicit solution system, combining the advantages of a diagonal mass matrix and the possibility of using unstructured meshes. The con- vergence rate has been tested in reference to the prob- lem of free harmonic vibrations in a system with one de- gree of freedom, showing that the Cell Method has the same convergence rate of II order Runge Kutta method, but its accuracy is better. The Cell Method results in 2D and 3D have been compared with those obtained with the commercial codes ANSYS and ABAQUS in the problem of the longitudinal vibration of a bar with free ends, for which the exact analytic solution is found in literature. The Cell Method results are comparable with or better than those obtained with FEM, and they are particularly interesting from the point of view of computation time and memory requirements for very large meshes. keyword: Numerical method, Cell method, Elasto- dynamics, Transient analysis 1 Introduction The modeling and simulation of systems behavior by means of numerical methods is a common procedure in the design and development phases of technological and industrial products. One of the most important aspects in the process of mechanical response evaluation con- sists in a correct estimate of the stress and strain state in the machine components under transient loading, partic- ularly when discontinuities and stress concentrations are present. The prediction of mechanical components be- havior by means of numerical simulations, performed on virtual components, accelerates the mechanical systems 1 Department of Mechanical Engineering, University of Trieste, Tri- este, Italy, cosmi@units.it optimization process and results in an important reduc- tion of experimental tests and project development costs. A widely used method is the Finite Element Method (FEM), either under the time domain or the frequency domain approach. In particular, transient analysis in time domain is computed by means of numerical integration. Two kinds of numerical integration methods are avail- able: conditionally stable methods and unconditionally stable methods. The first approach requires a small integration step and the answer of the system at the end of the integration step is computed based on the conditions at the beginning of the step. These methods, also called explicit meth- ods, are very convenient from the computational point of view but are only applicable when the mass matrix is diagonal. The other requirement is that the Courant con- dition be satisfied: the integration step must be smaller than the minimum period of time required for a distur- bance to travel between two nodes of the mesh, the wave propagation velocity in the material being known. Un- fortunately, the Finite Element Method mass matrix is in general not diagonal, so that explicit integration meth- ods are not employable. On the other side, the Finite Differences in Time Domain Method (FDTD) yields to a diagonal mass matrix and an explicit system but requires the use of structured meshes that present important draw- backs, for example difficulties in curve shapes modeling. Unconditionally stable methods, also called implicit methods, compute the answer of the system at the end of the integration step based on the conditions at the end of the step. The answer is numerically stable for any in- tegration step, but this doesn’t mean that any step can be used, because the solution accuracy decreases when the step becomes larger. The process is very heavy from the computational point of view, as it requires the solution of an algebraic system at each time step. The computational burden of a Finite Element dynamic analysis, necessarily higher than that of a static analysis, can be reduced if the mass matrix is rendered diagonal (lumping), at the cost