In the numerical analysis, the FEM belongs to a class of so called Galerkin methods which convert continuous problems to discrete problems. Originally, it was designed for solving the problems in structural mechanics mainly in construction of buildings.
Then, it became an excellent tool in many areas of modellings and simulations. As the deformation simulation method proposed in this thesis shares some FEM formulations with a discrete vibration system, the reader can refer to [Shabana,1990] for a detailed deducing process of FEM formulations. In this section, we will briefly present the discretization process of continuous solids, including the mesh generation, and the equation of motion.
2.2.1 Domain Discretization and Mesh Generation
In this process, the continuous domain Ω is discretized by a triangulation ΓΩ The triangulation is often referenced as finite element mesh. Each element is given by a set of nodes, edges and faces. The nodes are usually represented by the vertices of
the element. However, in the case of higher-order elements, supplemental nodes are added, e.g., being placed in the middle of the element edges or faces. In the case of two-dimensional domain, the most popular elements are represented by triangles and rectangles (see the (a), (b), (c) and the (d) in Fig.2.4), whereas tetrahedral, pyramidal, parallelepiped or prismatic elements are commonly used to discretize three-dimensional domains (see the (e) and (f ) in Fig. 2.4). A good triangulation of a complex domain is a non-trivial process, as the elements should fulfil some conditions concerning the shape. For example, taking tetrahedral elements into account, aspect ratio is computed as a measure of quality, which is usually defined as the maximum side length to the minimum altitude.
(a) (b)
(c) (d)
(e) (f)
Figure 2.4 – Several 2D and 3D elements used in FEM: (a) linear triangular element with 3 nodes, (b) linear rectangular element with 4 nodes, (c) quadratic triangular element 6 nodes, (d) Lagrangian element with 9 nodes, e tetrahedral element with 4 nodes, and (f ) brick element with 20 nodes.
Of course, the FEM provides several different types of elements. Throughout the thesis, we apply the first-order tetrahedron element (see the (e) in Fig. 2.4), which is the most common element type being used, because this is the simplest kind of a FEM element, with a constant value of strain inside each tetrahedron. The deformation modelling method proposed in this thesis will not change when other types of elements are considered. A particular body deformation is specified by the displacements of mesh vertices, while a small set of vertices are constrained to have zero displacements.
For example, for a volumetric mesh model consisting of n vertices, the displacement vector u ∈ IR3n contains the x, y, z world-coordinate displacements of model vertices.
As our goal application is to develop a framework for design validation of deformable mechanical parts, a mesh model with a sufficient meshing resolution is necessary to achieve an accurate deformation result. However, for haptic interactions with these deformation evaluations, the trade-off between the deformation accuracy, which is generally determined by the density of triangulation, and the real-time interaction performance has to be obtained. Regarding this point, we propose, in this thesis, an off-line mesh analysis method based on anticipated deformation validation scenar-ios. The method pre-computes different modal deformation spaces off-line and enables
the real-time deformation computation to be switched with respect to operators’ in-teractions. Our mesh analysis method can be viewed as an attempt to enhance the deformation accuracy on DOFs where are most important. Moreover, this method can be implemented regardless of the particular choice of the element type, although all our simulations through the thesis are based on the first order tetrahedron element.
2.2.2 The Equation of Motion
The unreduced equations of motion for structural vibrations of a volumetric 3D de-formable object, under the FEM discretization, can be derived from the principle of virtual work of Lagrangian mechanics. These equations of motion, normally denoted as the Euler-Lagrange equation, are a second-order system of ordinary differential equa-tions [Shabana, 1990].
M ¨u + D(u, ˙u) + R(u) = F (2.15) Here, u (u, v, w) ∈ IR3n is the displacement vector (the unknown), M ∈ IR3n,3n is the mass matrix, D(u, ˙u) are damping forces, and R(u) are internal deformation forces, and F ∈ IR3n is the external forces resulting from user interactions or collision reaction forces. The mass matrix is constant in time and depends only on the object’s mesh and mass density distribution in the undeformed configuration. In general, it is a sparse non-diagonal matrix, however for algorithmic convenience, it is often simplified into a diagonal matrix by accumulating all the row entries onto the diagonal element [Zhuang and Canny, 2000]. Such lumping essentially means that all elements re-assign their volumetrically distributed mass to their vertices: it is as if the model consisted of a point-like mass at every simulation vertex, with zero mass anywhere else inside the elements. Such a construction of course means losing some simulation accuracy. It is true that the accuracy loss is smaller with finer meshes.
For haptic applications, an explicit employment of the FEM leads to huge compu-tations and thus hinder the real-time interaction performance. One simplified method is based on small deformation assumption by employing the linear elasticity theory (see Equation 2.9 and 2.10). This linear strain makes the internal force vector linear with respect to nodal displacement vector. Namely, it simplifies Equation 2.15 to the following linear system:
M ¨u + D ˙u + Ku = F. (2.16)
As the stiffness matrix and the mass matrix are keeping constant during real-time deformation simulations, the above simplified linear system permits an efficient pre-computation process, which is a key to guarantee the real-time interaction performance.
As in the thesis, we focus on the deformation validation of complex industrial me-chanical parts, the dynamic solving of time-dependent differential equations is not suit-able for introducing haptic devices into these applications. In order to speed up the interactivity rate while keeping the accuracy advantages of the FEM models, we pro-pose a two-stage method combining an off-line pre-computation phase and an on-line deformation interaction phase. And our deformation modelling method takes advan-tage of the linearity of the equations to perform a pre-computation process based on the model reduction method described in section 3.2.3.
2.2.2.1 Special Cases
Sometimes, one is interested only in the deformations assumed under a certain fixed static load, as opposed to the dynamics of deformations. Such static simulations are useful, for example, when determining how structures are able to sustain applied loads.
In this section, we would like to point out two special cases of the equation of motion.
One special case is the system of equations for static deformations. It is obtained by setting the nodal accelerations and velocities to zeros. This leads to the following system of equations for static global deformation,
R(u) = F. (2.17)
Another special case is the corresponding linear system for static small deformation is written as,
Ku = F. (2.18)
In this thesis, our two-stage method based on the model reduction method can be applied equally well to these special cases concerning static deformation simulations:
simply discarding the reduced mass matrix and damping terms.