A Finite Element Method for Interactive Physically Based Shape Modelling with Quadratic Tetrahedra

We present an alternative approach to standard geometric shape editing using physically-based simulation. With our technique, the user can deform complex objects in real-time. The enabling technology of this approach is a fast and accurate finite element implementation of an elasto-plastic material model, specifically designed for interactive shape manipulation. Using quadratic shape functions, we avoid the inherent drawback of volume locking exhibited by methods based on linear finite elements. The physical simulation uses a tetrahedral mesh, which is constructed from a coarser approximation of the detailed surface. Having computed a deformed state of the tetrahedral mesh, the deformation is transferred back to the high detail surface. This can be accomplished in an accurate and efficient way using the quadratic shape functions. In order to guarantee stability and real-time frame rates during the simulation, we cast the elasto-plastic problem into a linear formulation. For this purpose, we present a corotational formulation for quadratic finite elements. We demonstrate the versatility of our approach in interactive manipulation sessions and show that our animation system can be coupled with further physics-based animations like, e.g. fluids and cloth, in a bi-directional way

Simulation of hyperelastic materials in real-time using Deep Learning

The finite element method (FEM) is among the most commonly used numerical methods for solving engineering problems. Due to its computational cost, various ideas have been introduced to reduce computation times, such as domain decomposition, parallel computing, adaptive meshing, and model order reduction. In this paper we present U-Mesh: a data-driven method based on a U-Net architecture that approximates the non-linear relation between a contact force and the displacement field computed by a FEM algorithm. We show that deep learning, one of the latest machine learning methods based on artificial neural networks, can enhance computational mechanics through its ability to encode highly non-linear models in a compact form. Our method is applied to two benchmark examples: a cantilever beam and an L-shape subject to moving punctual loads. A comparison between our method and proper orthogonal decomposition (POD) is done through the paper. The results show that U-Mesh can perform very fast simulations on various geometries, mesh resolutions and number of input forces with very small errors

Meshless Mechanics and Point-Based Visualization Methods for Surgical Simulations

Computer-based modeling and simulation practices have become an integral part of the medical education field. For surgical simulation applications, realistic constitutive modeling of soft tissue is considered to be one of the most challenging aspects of the problem, because biomechanical soft-tissue models need to reflect the correct elastic response, have to be efficient in order to run at interactive simulation rates, and be able to support operations such as cuts and sutures. Mesh-based solutions, where the connections between the individual degrees of freedom (DoF) are defined explicitly, have been the traditional choice to approach these problems. However, when the problem under investigation contains a discontinuity that disrupts the connectivity between the DoFs, the underlying mesh structure has to be reconfigured in order to handle the newly introduced discontinuity correctly. This reconfiguration for mesh-based techniques is typically called dynamic remeshing, and most of the time it causes the performance bottleneck in the simulation. In this dissertation, the efficiency of point-based meshless methods is investigated for both constitutive modeling of elastic soft tissues and visualization of simulation objects, where arbitrary discontinuities/cuts are applied to the objects in the context of surgical simulation. The point-based deformable object modeling problem is examined in three functional aspects: modeling continuous elastic deformations with, handling discontinuities in, and visualizing a point-based object. Algorithmic and implementation details of the presented techniques are discussed in the dissertation. The presented point-based techniques are implemented as separate components and integrated into the open-source software framework SOFA. The presented meshless continuum mechanics model of elastic tissue were verified by comparing it to the Hertzian non-adhesive frictionless contact theory. Virtual experiments were setup with a point-based deformable block and a rigid indenter, and force-displacement curves obtained from the virtual experiments were compared to the theoretical solutions. The meshless mechanics model of soft tissue and the integrated novel discontinuity treatment technique discussed in this dissertation allows handling cuts of arbitrary shape. The implemented enrichment technique not only modifies the internal mechanics of the soft tissue model, but also updates the point-based visual representation in an efficient way preventing the use of costly dynamic remeshing operations

Animation of deformable bodies with quadratic bézier finite elements

pre-printIn this article, we investigate the use of quadratic finite elements for graphical animation of deformable bodies.We consider both integrating quadratic elements with conventional linear elements to achieve a computationally efficient adaptive-degree simulation framework as well as wholly quadratic elements for the simulation of nonlinear rest shapes. In both cases, we adopt the B´ezier basis functions and employ a co-rotational linear strain formulation. As with linear elements, the co-rotational formulation allows us to precompute per-element stiffness matrices, resulting in substantial computational savings. We present several examples that demonstrate the advantages of quadratic elements in general and our adaptive-degree system in particular. Furthermore, we demonstrate, for the first time in computer graphics, animations of volumetric deformable bodies with nonlinear rest shapes

Isogeometric analysis : applications for torque and drag models, drillstring and bottom-hole assembly design

The drilling industry today relies on torque and drag models to analyze and ensure success during all phases of well construction and operations, including planning, drilling, and completion. Analytical models are based on equations that are undergoing constant development and improvement. The finite element method is an alternative to complex analytical calculations that is used often to determine torque and drag forces that are present when a drillstring is lowered, raised, and rotated in a wellbore. Traditional finite element analysis (FEA), however, is not time efficient or computationally able to simulate the complexities of a real wellbore. Thus, we introduce an alternative to the traditional finite element approach: isogeometric analysis. Isogeometric analysis is similar to finite element analysis except that it uses NURBS (Non-Uniform Rational B-Splines), as opposed to interpolatory polynomials used in traditional FEA, as the basis functions. NURBS functions are the same as those used in CAD programs, and they are able to construct exact conic shapes, such as circles and ellipses. Adopting NURBS basis functions allows finite element analysis to be performed directly on the exact geometrical surface - not on an approximate geometric surface mesh, as in traditional FEA. IGA yields a significantly faster and more accurate simulation. This thesis presents a real-world application of IGA to a drag force model to determine the resultant surface hook load during run-in-hole (RIH) operations. Real well data is used, and IGA results are compared to a similar FEA analysis. The outcome shows that IGA is indeed a superior finite element method that has immense potential for further application in the industryPetroleum and Geosystems Engineerin

Phase-field modeling of brittle fracture with multi-level hp-FEM and the finite cell method

The difficulties in dealing with discontinuities related to a sharp crack are overcome in the phase-field approach for fracture by modeling the crack as a diffusive object being described by a continuous field having high gradients. The discrete crack limit case is approached for a small length-scale parameter that controls the width of the transition region between the fully broken and the undamaged phases. From a computational standpoint, this necessitates fine meshes, at least locally, in order to accurately resolve the phase-field profile. In the classical approach, phase-field models are computed on a fixed mesh that is a priori refined in the areas where the crack is expected to propagate. This on the other hand curbs the convenience of using phase-field models for unknown crack paths and its ability to handle complex crack propagation patterns. In this work, we overcome this issue by employing the multi-level hp-refinement technique that enables a dynamically changing mesh which in turn allows the refinement to remain local at singularities and high gradients without problems of hanging nodes. Yet, in case of complex geometries, mesh generation and in particular local refinement becomes non-trivial. We address this issue by integrating a two-dimensional phase-field framework for brittle fracture with the finite cell method (FCM). The FCM based on high-order finite elements is a non-geometry-conforming discretization technique wherein the physical domain is embedded into a larger fictitious domain of simple geometry that can be easily discretized. This facilitates mesh generation for complex geometries and supports local refinement. Numerical examples including a comparison to a validation experiment illustrate the applicability of the multi-level hp-refinement and the FCM in the context of phase-field simulations

ElVis: A system for the accurate and interactive visualization of high-order finite element solutions

pre-printThis paper presents the Element Visualizer (ElVis), a new, open-source scientific visualization system for use with high order finite element solutions to PDEs in three dimensions. This system is designed to minimize visualization errors of these types of fields by querying the underlying finite element basis functions (e.g., high-order polynomials) directly, leading to pixel-exact representations of solutions and geometry. The system interacts with simulation data through run time plugins, which only require users to implement a handful of operations fundamental to finite element solvers. The data in turn can be visualized through the use of cut surfaces, contours, isosurfaces, and volume rendering. These visualization algorithms are implemented using NVIDIA's OptiX GPU-based ray-tracing engine, which provides accelerated ray traversal of the high-order geometry, and CUDA, which allows for effective parallel evaluation of the visualization algorithms. The direct interface between ElVis and the underlying data differentiates it from existing visualization tools. Current tools assume the underlying data is composed of linear primitives; high-order data must be interpolated with linear functions as a result. In this work, examples drawn from aerodynamic simulations-high-order discontinuous Galerkin finite element solutions of aerodynamic flows in particular-will demonstrate the superiority of ElVis' pixel-exact approach when compared with traditional linear-interpolation methods. Such methods can introduce a number of inaccuracies in the resulting visualization, making it unclear if visual artifacts are genuine to the solution data or if these artifacts are the result of interpolation errors. Linear methods additionally cannot properly visualize curved geometries (elements or boundaries) which can greatly inhibit developers' debugging efforts. As we will show, pixel-exact visualization exhibits none of these issues, removing the visualization scheme as a source of uncertainty for engineers using ElVis