Locking-Proof Tetrahedra

The simulation of incompressible materials suffers from locking when using the standard finite element method (FEM) and coarse linear tetrahedral meshes. Locking increases as the Poisson ratio gets close to 0.5 and often lower Poisson ratio values are used to reduce locking, affecting volume preservation. We propose a novel mixed FEM approach to simulating incompressible solids that alleviates the locking problem for tetrahedra. Our method uses linear shape functions for both displacements and pressure, and adds one scalar per node. It can accommodate nonlinear isotropic materials described by a Young\u27s modulus and any Poisson ratio value by enforcing a volumetric constitutive law. The most realistic such material is Neo-Hookean, and we focus on adapting it to our method. For , we can obtain full volume preservation up to any desired numerical accuracy. We show that standard Neo-Hookean simulations using tetrahedra are often locking, which, in turn, affects accuracy. We show that our method gives better results and that our Newton solver is more robust. As an alternative, we propose a dual ascent solver that is simple and has a good convergence rate. We validate these results using numerical experiments and quantitative analysis

Analysis of the compressible, isotropic, neo-Hookean hyperelastic model

The most widely-used representation of the compressible, isotropic, neo-Hookean hyperelastic model is considered in this paper. The version under investigation is that which is implemented in the commercial finite element software ABAQUS, ANSYS and COMSOL. Transverse stretch solutions are obtained for the following homogeneous deformations: uniaxial loading, equibiaxial loading in plane stress, and uniaxial loading in plane strain. The ground-state Poisson’s ratio is used to parameterize the constitutive model, and stress solutions are computed numerically for the physically permitted range of its values. Despite its broad application to a number of engineering problems, the physical limitations of the model, particularly in the small to moderate stretch regimes, are not explored. In this work, we describe and analyze results and make some critical observations, underlining the model’s advantages and limitations. For example, a snap-back feature of the transverse stretch is identified in uniaxial compression, a physically undesirable behavior unless validated by experimental data. The domain of this non-unique solution is determined in terms of the ground-state Poisson’s ratio and the state of stretch and stress. The analyses we perform are essential to enable the understanding of the characteristics of the standard, compressible, isotropic, neo-Hookean model used in ABAQUS, ANSYS and COMSOL. In addition, our results provide a framework for the parameter-fitting procedure needed to characterize this standard, compressible, isotropic neo-Hookean model in terms of experimental data

Towards Real-Time Simulation Of Hyperelastic Materials

We propose a new method for physics-based simulation supporting many different types of hyperelastic materials from mass-spring systems to three-dimensional finite element models, pushing the performance of the simulation towards real-time. Fast simulation methods such as Position Based Dynamics exist, but support only limited selection of materials; even classical materials such as corotated linear elasticity and Neo-Hookean elasticity are not supported. Simulation of these types of materials currently relies on Newton\u27s method, which is slow, even with only one iteration per timestep. In this work, we start from simple material models such as mass-spring systems or as-rigid-as-possible materials. We express the widely used implicit Euler time integration as an energy minimization problem and introduce auxiliary projection variables as extra unknowns. After our reformulation, the minimization problem becomes linear in the node positions, while all the non-linear terms are isolated in individual elements. We then extend this idea to efficiently simulate a more general spatial discretization using finite element method. We show that our reformulation can be interpreted as a quasi-Newton method. This insight enables very efficient simulation of a large class of hyperelastic materials. The quasi-Newton interpretation also allows us to leverage ideas from numerical optimization. In particular, we show that our solver can be further accelerated using L-BFGS updates (Limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm). Our final method is typically more than ten times faster than one iteration of Newton\u27s method without compromising quality. In fact, our result is often more accurate than the result obtained with one iteration of Newton\u27s method. Our method is also easier to implement, implying reduced software development costs

Instabilities in the axisymmetric magnetoelastic deformation of a cylindrical membrane

We study the inflation of a weakly magnetizable isotropic incompressible cylindrical membrane and the effects of an external magnetic field generated by a current carrying wire placed along the axis of cylinder. A variational formulation based on magnetization is used and the computational results obtained by using four elastic constitutive models (neo-Hookean, Mooney–Rivlin, Ogden, and Arruda–Boyce) are studied and compared. Cylinders of various aspect ratios are studied in each case. Our study shows that the external magnetic field alters the elastic limit point, does not lead to equilibrium solutions below certain value of internal pressure, and can give rise to multiple equilibrium states for a given value of pressure. Presence of magnetic limit point, a phenomenon recently reported in the literature is reconfirmed. Magnetic limit point is a state where a further strengthening of the applied magnetic field at a given pressure does not yield any static equilibrium state. In this case it is detected when the cylindrical membrane deflates into the volume enclosed by itself. We also observe a quadratic relation between the defined magnetic energy parameter and the internal pressure at the magnetic limit point. Relaxed form of the strain energy density is used to account for wrinkling in this case of inward inflation. A finite difference method coupled with an arc-length technique is used for the computations and the stability of the solution is determined from the second variation

Fast soft-tissue deformations with FEM

Soft body simulation has been a very active research area in computer animation since Baraff and Witkin's 1998 work on cloth simulation, which led Pixar to start using such techniques in all of its animated movies that followed. Many challenges in these simulations come from different roots. From a numerical point of view, deformable systems are large sparse problems that can become numerically unstable at surprising rates and may need to be modified at each time-step. From a mathematical point of view, hyperelastic models defined by continuum mechanics need to be derived, established and configured. And from the geometric side, physical interaction with the environment and self-collisions may need to be detected and introduced into the solver. It is a fact that the Computer Graphics academia primarily focuses on offline methods, both for rendering and simulation. At the same time, the advances from the industry mainly apply to real-time rendering. However, we wondered how such high-quality simulation methods would map to a real-time use case. In this thesis, we delve into the simulation system used by Pixar's Fizt2 simulator, based on the Finite Element Method, and investigate how to apply the same techniques in real-time while preserving robustness and fidelity, altogether providing the user with some interaction mechanisms. A 3D engine for simulating deformable materials has been developed following the described models, with an interactive interface that allows the definition and configuration of scenes and later interaction with the simulation

An hourglass-free formulation for total Lagrangian smoothed particle hydrodynamics

The total Lagrangian smoothed particle hydrodynamics (TL-SPH) for elastic solid dynamics suffers from hourglass modes which can grow and lead to the failure of simulation for problems with large deformation. To address this long-standing issue, we present an hourglass-free formulation based on volumetric-devioatric stress decomposition. Inspired by the fact that the artifact of nonphysical zigzag particle distribution induced by the hourglass modes is mainly characterized by shear deformation and the standard SPH discretization for the viscous term in the Navier-Stokes (NS) equation, the present formulation computes the action of shear stress directly through the Laplacian of displacement other than from the divergence of shear stress. A comprehensive set of challenging benchmark cases are simulated to demonstrate that, while improving accuracy and computational efficiency, the present formulation is able to eliminate the hourglass modes and achieves very good numerical stability with a single general effective parameter. In addition, the deformation of a practically relevant stent structure is simulated to demonstrate the potential of the present method in the field of biomechanics.Comment: 38 pages 21 figure

Experimental and numerical investigation of soft impact loading on aircraft materials

Bird strike poses a great hazard to aircraft components, such as the engine and windshield, during flight. Thus, it is critical to understand the impact resistance of key aircraft components under bird strike. During this PhD study, Aluminium Alloy 2024-T3 and laminated glass interlayered with thermoplastic polyurethane (TPU), which are commonly used as the materials of the aircraft fuselage and windshield respectively, are selected as impact target materials. Both laboratory-based experiments and finite element simulations are performed using a light gas gun and ABAQUS/EXPLICIT respectively. A good agreement is achieved between the experimental work and numerical predictions for the impact response. Two bird substitute materials were selected: RTV rubber and ballistic gelatine, whose dynamic pressure profiles are similar to that of a real bird during a high speed impact. Mechanical properties of both materials were investigated by conducting compression tests at quasi-static (0.25, 2.5 and 25 min-1) and intermediate rate (2760 min-1 to 22500 min-1). As a result, the relationship between true stress and strain is obtained and constitutive equations are established using the hyperelastic models, i.e. the Ogden, Mooney-Rivlin and Neo-Hookean. The 3D Digital Image Correlation technique was employed in the gas gun test of the AA 2024-T3 and TPU interlayered laminated glass. Thus, the impact compliance of both targets from rubber and gelatine impacts are attained and found to be roughly equal to the same initial projectile momentum. An FE simulation was used to model the experimental process and was validated against the DIC results. Moreover, the Hugoniot pressure plays a predominant role in laminated glass fracture during the impact, with the rubber projectile leading to a larger damage than the gelatine projectile given the same momentum. This is because the shock wave speed in the rubber is larger than that in the gelatine projectile. A validated FE simulation is therefore presented, which can be used to simulate real bird impact on real aircraft structures in the future. Thus, this PhD work can be potentially implemented into industrial research programmes to aid design and optimisation.Open Acces