Deformable 3D Objects for a VR medical application M. Mero Dpto. de Matem´ aticas - FACYT, Universidad de Carabobo (Venezuela) mmero@lsi.upc.es A. Susin Dpt. Matem` atica Aplicada 1 Universitat Polit` ecnica de Catalunya (Spain) toni.susin@upc.es Abstract In this paper , we enhance existing techniques for simulating flexible volumetric objects. The idea is to use a mixed model of Finite Element and Mesh Free Methods. From this approach we will be able to build a 3D deformable model which can be included in a gene- ral application of virtual reality for a medical surgery simulator. The final model will be the heart of a patient builded from their actual SPECT data. The mixed ap- proach allow us to construct a multiresolution model that can be used to obtain real time response when an extern user interacts with the model. 1 Introduction Simulating and animating 3D deformable objects in real time is essential to many interactive applications such as surgery simulators. One of the main charac- teristics of these simulations is the dynamic interac- tion between the deformable model and the possible external forces acting on it. The dynamic behavior of our volumetric 3D deformable model is based on linear elastic mechanics. It is essentially based on techniques presented in computer graphics and mechanical engi- neering literature [RD89] , [Y.F65] and [D.T86]. From the work developed by J. O’Brien and J. Hod- gins [JJ99] and G. Debunne [GMMA01] we can state the formulation of the problem in terms of Finite Ele- ments (FEM). Although [JJ99] is focused on the study of fractures in almost rigid materials, the elastic model can be used for simulating more deformable objects. In particular, [GMMA01] build a model for the human liver with the same finite element formulation. When the final goal of one application is a Virtual Reality environment, topics like precision and speed need to be properly balanced. To combine both cha- racteristics the usual solution is to build a multiresolu- tion model [GMMA01]. The structure of such a model is organized in different layers from a coarse to a fi- ner mesh. The computational accuracy is related to the number and size of the elements. When an exter- nal force is applied to the model in a delimited zone, the finer mesh is activated. The other resolutions are used to animate the model according to the distance from the force location. The different multiresolution models differs in how are related two consecutive mesh levels. They can be obtained from a refinement of the previous level mesh or completely independent (just meshing the same volume). This relation is critical in the transition zone where the two different meshes have to be activated. In any case, we are forced to work with many mes- hes with the resulting increase in the difficulty of the data structure of the problem. One possibility to avoid this problem is to use only one mesh (i.e. a coarser mesh) and apply another multiresolution approach to the deformable model. The idea is to refine the zone where the forces are applied with particles and to use a Mesh Free Method (MFM) for computing the elastic reaction of the model ([L.L77],[JCCW96],[WYS + 96], [AS00],[S.F01]). In this way, our idea is to build a mixed model with FME and MFM that simplify the data structure and uses the multiresoltion approach. A transition zone is defined in order to achieve this goal. 2 Finite Element Formulation The usual modelization is based on continuum me- chanics [Fung]. Regarding our simulation, the first as- sumption in the continuum approach defines the scale effects is significantly greater than the scale of the ma- terial’s composition. Therefore, the behavior of the molecules or particles that compose of the material can be modeled as continuous media. We begin the description of the continuous model by defining material coordinates that parameterize the volume of space occupied by the object being modeled.