Homogeneous finite-element and surface construction method for stable real-time cutting and haptic simulation

Abstract

Real-time interactive simulations based on the virtual environments provide users with a variety of realistic experiences. Immediate visual and force feedbacks are necessary according to the user inputs, and each feedback requires a fast computation with a refresh rate of 30 Hz and 300 – 1,000 Hz, respectively. Dynamic simulation of the deformable models requires lots of computations. Computational complexity for this simulation is typically known as $O(n^{3})$, where n is the degrees of freedom of the model. Despite of the great improvements in the processing hardware, it is still difficult to achieve a high fresh rate for the force rendering, i.e., 300 - 1,000 Hz. Furthermore, the difficulty is further exacerbated in the cutting or fracture simulation due to additional computation burdens and numerical instability problems. Previous approaches such as multi-rate methods, learning-based methods, and order reduction methods address this problem. In this dissertation, the deformable model consists of a single type of regular hexahedrons only. Because the object geometry is not filled with regular hexahedrons only, small gaps between the object boundaries and the hexahedrons are allowed. Additional surface structure is constructed for surface representation, and is approximated from the regular hexahedrons. Features of this modeling method are exploited in the deformation and cutting simulation method. Firstly, parallel computing method of computing the model deformation is introduced. The entire computation is formulated into a node-wise form, taking into account that the element stiffness matrices are the same and the number of hexahedrons around the node is small. Methods to simulate large deformations and anisotropic material properties in the model are also presented. The model boundaries are represented by a triangular mesh, which is approximated by neighboring hexahedral nodes and Moving-least-squares (MLS) approximation functions. A method of grouping the neighboring nodes and computing the corresponding MLS functions is introduced. Two separate ways of updating the surface meshes are introduced

the entire mesh is updated for each visual loop, whereas a few meshes around the tool are locally updated for each haptic loop. This approach dramatically reduces the computational burden for a haptic loop. Second, cutting simulation method is proposed, which faithfully represents a given cutting path but still maintaining the numerical stability condition. The proposed method duplicates the intersected hexahedrons, so the reference size and shape can be maintained. This approach ensures numerical stability condition of the entire model based on the relevant theoretical background. Additional surface meshes and vertices are created to faithfully represent the cutting path. It is important to determine the group of neighboring nodes for each surface vertex, because the two different sides of the cut surface must be properly separated. Besides, the group of surface vertices to be updated should be minimized for the computational efficiency. Methods to deal with such problems are introduced. A few cutting cases including a partial cutting of the model, a curved cutting path, and a concave-shaped model are simulated. Cutting of soft tissues using a surgical knife involves repeated loadings (or deformations) and sudden force drops (or cuttings). This phenomenon is reflected in this dissertation to render a realistic force feedback during the cutting. In the loading section, each contact vertex is mapped to the closest hexahedral node to share the penetration depth and the contact force. Cutting is simulated when the contact force exceeds a threshold. Since the cut surface is generated within the model, the contact force is reduced in the next simulation time step. The computational efficiency of the proposed method is compared with previous methods in the literature. Simulation results show that the proposed deformation simulation method is approximately 10 times faster than the previous methods. The proposed method can simulate the deformation of a model consisting of 32,712 nodes with a refresh rate of more than 500 Hz using a typical PC. Besides, additional computation burdens from the cutting simulation are compared with the previous method. The theoretical range of additional nodes and elements is derived, and the average computation time is compared with a few cutting simulation examples. Both theoretical and simulation results show that the additional computation burden of the proposed method is reduced to about 30 % compared to the previous method. Small gaps between the given object boundaries and the hexahedrons deteriorate the simulation accuracy. This problem is exacerbated as the object size becomes smaller compared to the hexahedron size. This dissertation introduces a method of considering a partial element, which is partially included inside of the object geometry. Additional computations required for the partial elements and the corresponding error reduction are presented. The model constructed in the proposed method is compared with the conventional Finite-element Method (FEM) model, and the differences are verified in terms of the human Just Noticeable Difference (JND).