Corotated meshless implicit dynamics for deformable bodies
Stephane Cotin2019, Computer Science Research Notes
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
This paper proposes a fast, stable and accurate meshless method to simulate geometrically non-linear elastic behaviors. To address the inherent limitations of finite element (FE) models, the discretization of the domain is simplified by removing the need to create polyhedral elements. The volumetric locking effect exhibited by incompressible materials in some linear FE models is also completely avoided. Our approach merely requires that the volume of the object be filled with a cloud of points. To minimize numerical errors, we construct a corotational formulation around the quadrature positions that is well suited for large displacements containing small deformations. The equations of motion are integrated in time following an implicit scheme. The convergence rate and accuracy are validated through both stretching and bending case studies. Finally, results are presented using a set of examples that show how we can easily build a realistic physical model of various deformable bodies with little effort spent on the discretization of the domain.
Related papers
Volume conserving finite element simulations of deformable models
ACM Transactions on Graphics, 2007
We propose a numerical method for modeling highly deformable nonlinear incompressible solids that conserves the volume locally near each node in a finite element simulation. Our method works with arbitrary constitutive models, is applicable to both passive materials and active materials (e.g. muscles), and works with simple tetrahedra without the need for multiple quadrature points or stabilization techniques. Although simple linear tetrahedra typically suffer from locking when modeling incompressible materials, our method enforces incompressibility per node (in a one-ring), and we demonstrate that it is free from locking. Finally, we propose a novel method for treating both collisions and self-collisions as linear constraints during the incompressible solve, alleviating issues in enforcing multiple possibly conflicting constraints.
Real-time meshless deformation
Computer Animation and Virtual Worlds, 2005
In this paper, we articulate a meshless computational paradigm for the effective modeling, accurate physical simulation, and real-time animation of point-sampled solid objects. Both the interior and the boundary geometry of our volumetric object representation only consist of points, further extending the powerful and popular method of point-sampled surfaces to the volumetric setting. We build the point-based physical model upon continuum mechanics, which affords to effectively model the dynamic elastic behavior of pointbased volumetric objects. When only surface samples are provided, our prototype system first generates both interior volumetric points and a volumetric distance field with octree structure. The physics of these volumetric points in a solid interior are simulated using the Meshless Moving Least Squares (MLS) shape functions. In sharp contrast to the traditional finite element method (FEM), the meshless property of our new technique expedites the accurate representation and precise simulation of the underlying discrete model, without the need of domain meshing. In order to achieve real-time simulations, we utilize the warped modal analysis method that is locally linear in nature but globally warped to account for rotational deformation. The structural simplicity and real-time performance of our meshless simulation framework are ideal for interactive animation and game/movie production.
Galerkin meshfree methods applied to the nonlinear dynamics of flexible multibody systems
Multibody System Dynamics, 2011
Meshfree Galerkin methods have been developed recently for the simulation of complex mechanical problems involving large strains of structures, crack propagation or high velocity impact dynamics. At the present time, the application of these methods to multibody dynamics has not been made despite their great advantage in some situations over standard finite element techniques.
Meshfree elastoplastic solid for non-smooth multi-domain dynamics
arXiv: Soft Condensed Matter, 2015
A method for simulation of elastoplastic solids in multibody systems with nonsmooth and multidomain dynamics is developed. The solid is discretised into pseudo-particles using the meshfree moving l ...
Sparse meshless models of complex deformable solids
ACM Transactions on Graphics, 2011
Meshless Methods: a Review and Computer Implementation Aspects
Mathematics and Computers …, 2008
Highly efficient FE simulations by means of simplified corotational formulation
Operational Research in Engineering Sciences: Theory and Applications, 2020
Finite Element Method (FEM) has deservedly gained the reputation of the most powerful numerical method in the field of structural analysis. It offers tools to perform various kinds of simulations in this field, ranging from static linear to nonlinear dynamic analyses. In recent years, a particular challenge is development of FE formulations that enable highly efficient simulations, aiming at real-time dynamic simulations as a final objective while keeping high simulation fidelity such as nonlinear effects. The authors of this paper propose a simplified corotational FE formulation as a possible solution to this challenge. The basic idea is to keep the linear behavior of each element in the FE assemblage, but to extract the rigid-body motion on the element level and include it in the formulation to cover geometric nonlinearities. This paper elaborates the idea and demonstrates it on static cases with three different finite element types. The objective is to check the achievable accuracy based on such a simplified geometrically nonlinear FE formulation. In the considered examples, the difference between the results obtained with the present formulation and those by rigorous formulations is less than 3% although fairly large deformations are induced.
Speeding up the simulation of deformable objects through mesh improvement
Computer Animation and Virtual Worlds, 2012
We propose a novel mesh optimization approach that is useful for speeding up a simulation of finite elements-based deformable objects. The approach is based on a quality metric derived from the critical simulation time step of explicit time integration schemes (i.e., the stability limit for the integration of dynamic equations). Our mesh smoothing approach consists of a set of small and independent spring systems. These are made up of a reference mesh node connected to a set of fixed endpoints, which represent the positions that maximize the time step of the elements adjacent to that node. The reference node is displaced to an equilibrium position through a few local iterations. Each spring's stiffness is weighted depending on the quality of its corresponding element. All spring systems can be computed in parallel. Global iterations update the mesh and spring systems. In addition, we combine our smoothing algorithm with topological transformations. With this approach, the simulation performance could be increased by more than 30% depending on the mesh. This approach is suitable for the generation of finite element method meshes, particularly those requiring interactive applications and haptic rendering.
A simple geometric model for elastic deformations
ACM Transactions on Graphics, 2010
We advocate a simple geometric model for elasticity: distance between the differential of a deformation and the rotation group. It comes with rigorous differential geometric underpinnings, both smooth and discrete, and is computationally almost as simple and efficient as linear elasticity. Owing to its geometric non-linearity, though, it does not suffer from the usual linearization artifacts. A material model with standard elastic moduli (Lamé parameters) falls out naturally, and a minimizer for static problems is easily augmented to construct a fully variational 2nd order time integrator. It has excellent conservation properties even for very coarse simulations, making it very robust. Our analysis was motivated by a number of heuristic, physics-like algorithms from geometry processing (editing, morphing, parameterization, and simulation). Starting with a continuous energy formulation and taking the underlying geometry into account, we simplify and accelerate these algorithms while avoiding common pitfalls. Through the connection with the Biot strain of mechanics, the intuition of previous work that these ideas are "like" elasticity is shown to be spot on.
A Discrete Approach to Meshless Lagrangian Solid Modeling
Computation
The author demonstrates a stable Lagrangian solid modeling method, tracking the interactions of solid mass particles rather than using a meshed grid. This numerical method avoids the problem of tensile instability often seen with smooth particle applied mechanics by having the solid particles apply stresses expected with Hooke's law, as opposed to using a smoothing function for neighboring solid particles. This method has been tested successfully with a bar in tension, compression, and shear, as well as a disk compressed into a flat plate, and the numerical model consistently matched the analytical Hooke's law as well as Hertz contact theory for all examples. The solid modeling numerical method was then built into a 2-D model of a pressure vessel, which was tested with liquid water particles under pressure and simulated with smoothed particle hydrodynamics. This simulation was stable, and demonstrated the feasibility of Lagrangian specification modeling for fluid-solid interactions.
Related papers
A multigrid framework for real-time simulation of deformable bodies
Computers & Graphics, 2006
In this paper, we present a multigrid framework for constructing implicit, yet interactive solvers for the governing equations of motion of deformable volumetric bodies. We have integrated linearized, corotational linearized and non-linear Green strain into this framework. Based on a 3D finite element hierarchy, this approach enables realistic simulation of objects exhibiting an elastic modulus with a dynamic range of several orders of magnitude. Using the linearized strain measure, we can simulate 50 thousand tetrahedral elements with 20 fps on a single processor CPU. By using corotational linearized and non-linear Green strain, we can still simulate five thousand and two thousand elements, respectively, at the same rates.
A Systematic Approach to the Element-Independent Corotational Dynamics of Finite Elements
The corotational (CR) kinematic description is the most recent of the formulations of geometrically nonlinear structural analysis. Because of this newness, it has not reached the same level of maturity of the older Lagrangian formulations: Total and Updated. Much work remains to be done, particularly in dynamics, nonconservative effects, and coupled field system applications. We are presently using this formulation for the structural component of an ambitious program for the coupled-field simulation (considering fluid-structure and control-structure interactions) of flight of complete flexible aircraft, including emergency maneuvers as in combat situations. The element-independent version of the CR description has the key advantage of allowing reuse of existing high-performance linear finite elements, in particular shells and solids. This paper provides the necessary element-independent mathematical tools to formulate the dynamic equations of motion in a moving, best-fit corotational frame including nonconservative effects.
Suite of meshless algorithms for accurate computation of soft tissue deformation for surgical simulation
Medical Image Analysis, 2019
The ability to predict patient-specific soft tissue deformations is key for computer-integrated surgery systems and the core enabling technology for a new era of personalized medicine. Element-Free Galerkin (EFG) methods are better suited for solving soft tissue deformation problems than the finite element method (FEM) due to their capability of handling large deformation while also eliminating the necessity of creating a complex predefined mesh. Nevertheless, meshless methods based on EFG formulation, exhibit three major limitations: i) meshless shape functions using higher order basis cannot always be computed for arbitrarily distributed nodes (irregular node placement is crucial for facilitating automated discretization of complex geometries); ii) imposition of the Essential Boundary Conditions (EBC) is not straightforward; and, iii) numerical (Gauss) integration in space is not exact as meshless shape functions are not polynomial. This paper presents a suite of Meshless Total Lagrangian Explicit Dynamics (MTLED) algorithms incorporating a Modified Moving Least Squares (MMLS) method for interpolating scattered data both for visualization and for numerical computations of soft tissue deformation, a novel way of imposing EBC for explicit time integration, and an adaptive numerical integration procedure within the Meshless Total Lagrangian Explicit Dynamics algorithm. The appropriateness and effectiveness of the proposed methods is demonstrated using comparisons with the established non-linear procedures from commercial finite element software ABAQUS and experiments with very large deformations. To demonstrate the translational benefits of MTLED we also present a realistic brain-shift computation.
Corotated Finite Elements Made Fast and Stable
Multigrid finite-element solvers using the corotational formulation of finite elements provide an attractive means for the simulation of deformable bodies exhibiting linear elastic response. The separation of rigid body motions from the total element motions using purely geometric methods or polar decomposition of the deformation gradient, however, can introduce instabilities for large element rotations and deformations. Furthermore, the integration of the corotational formulation into dynamic multigrid elasticity simulations requires to continually rebuild consistent system matrices at different resolution levels. The computational load imposed by these updates prohibits the use of large numbers of finite elements at rates comparable to the small-strain finite element formulation. To overcome the first problem, we present a new method to extract the rigid body motion from total finite element displacements based on energy minimization. This results in a very stable corotational formulation that only slightly increases the computational overhead. We address the second problem by introducing a novel algorithm for computing sparse products of the form R KR T , as they have to be evaluated to update the multigrid hierarchy. By reformulating the problem into the simultaneous processing of a sequential data and control stream, cache miss penalties are significantly reduced. Even though the algorithm increases memory requirements, it accelerates the multigrid FE simulation by a factor of up to 4 compared to previous multigrid approaches. Due to the proposed improvements, finite element deformable body simulations using the corotational formulation can be performed at rates of 17 tps for up to 12k elements.
Total Lagrangian explicit dynamics finite element algorithm for computing soft tissue deformation
Communications in Numerical Methods in Engineering, 2007
We propose an efficient numerical algorithm for computing deformations of 'very' soft tissues (such as the brain, liver, kidney etc.), with applications to real-time surgical simulation. The algorithm is based on the finite element method using the total Lagrangian formulation, where stresses and strains are measured with respect to the original configuration. This choice allows for pre-computing of most spatial derivatives before the commencement of the time-stepping procedure.
A Multigrid Framework for Real-Time Simulation of Deformable Volumes
2005
In this paper, we present a multigrid framework for constructing implicit, yet interactive solvers for the governing equations of motion of deformable volumetric bodies. We have integrated linearized, corotational linearized and non-linear Green strain into this framework. Based on a 3D finite element hierarchy, this approach enables realistic simulation of objects exhibiting an elastic modulus with a dynamic range of several orders of magnitude. Using the linearized strain measure, we can simulate 50 thousand tetrahedral elements with 20 fps on a single processor CPU. By using corotational linearized and non-linear Green strain, we can still simulate five thousand and two thousand elements, respectively, at the same rates.
An Adaptive Meshless Method for Modeling Large Mechanical Deformation and Contacts
Proceedings 2007 IEEE International Conference on Robotics and Automation, 2007
Growing new robotic applications in agriculture, food-processing, assisted surgery and haptics, which requires handling of highly deformable objects, present a number of design challenges; among these are methods to analyze deformable contacts. Recently, meshless methods (MLM), which inherit many advantages of finite element method (FEM) and yet need no explicit mesh structure to discretize geometry, have been proposed as an attractive alternative to FEM for solving engineering problems where automatic re-meshing is needed. This paper offers an adaptive MLM (automatically inserting nodes into large error regions) for solving contact problems. We employ the sliding line algorithm with the penalty method to handle contact constraints; it does not rely on small displacement assumptions and thus, it can solve non-linear contact problems with large deformation. Along with three practical applications, we validate the method against results computed using commercial FEM software and analytical solutions.
Improved Meshless Deformation Techniques for Plausible Interactive Soft Object Simulations
Meshless deformation based on shape matching is a new technique for simulating deformable objects without requiring mesh connectivity information. The approach focuses on speed, ease of use and stability at the expense of physical accuracy. In this paper we introduce improvements to the technique that increase physical realism and make it more suitable for use in interactive real-time environments such
U–W Formulation for Dynamic Problems in Large Deformation Regime Solved Through an Implicit Meshfree Scheme
Computational Mechanics
Solving dynamic problems for fluid saturated porous media at large deformation regime is an interesting but complex issue. An implicit time integration scheme is herein developed within the framework of the u − w (solid displacement-relative fluid displacement) formulation for the Biot's equations. In particular, liquid water saturated porous media is considered and the linearization of the linear momentum equations taking into account all the inertia terms for both solid and fluid phases is for the first time presented. The spatial discretization is carried out through a meshfree method, in which the shape functions are based on the principle of local maximum entropy LME. The current methodology is firstly validated with the dynamic consolidation of a soil column and the plastic shear band formulation of a square domain loaded by a rigid footing. The feasibility of this new numerical approach for solving large deformation dynamic problems is finally demonstrated through the application to an embankment problem subjected to an earthquake.
Meshless approach as an alternative to finite element method in solid mechanics numerical modeling
Rad Hrvatske akademije znanosti i umjetnosti. Tehničke znanosti, 2018
Meshless approaches enable discretizations of a computational model only by a set of nodes, which do not need to be connected to elements. This paper presents the meshless local Petrov-Galerkin method, which belongs to truly meshless approaches, as it does not require any kind of mesh or background cells for either interpolation or integration. Full displacement and mixed formulations are presented. The full displacement approach is used for the solution of a three-dimensional elasto-static problem, while the mixed approach is applied for the modeling of deformation responses of shell-like structures. The modeling of material discontinuities is performed by the mixed meshless local Petrov-Galerkin approach by employing the collocation method. The efficiency and accuracy of all the presented methods are tested and compared with finite element formulations in numerical examples. It is demonstrated that the meshless approaches may be considered an alternative to the well-known finite element method regarding certain problems.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.