Hourglass Modes Research Articles

Unified non-hourglass formulation for total Lagrangian SPH solid dynamics

AbstractThe persistence of hourglass modes poses a significant numerical instability issue in total Lagrangian smoothed particle hydrodynamics (TLSPH) solid dynamics, especially when dealing with substantial deformations, regardless of material properties. However, existing hourglass control methods have shown effectiveness only within limited applications. Thus far, a comprehensive solution capable of addressing hourglass issues across a wide range of material models, including elasticity, plasticity, and anisotropy, remains elusive. In this study, we introduce a unified TLSPH formulation grounded in volumetric-deviatoric stress decomposition, aimed at fundamentally mitigating hourglass modes in general simulations. Different conceptually from previous approaches using stress points or extra viscous or hourglass-control stresses within the momentum equation, our formulation is based on the weighted average of a standard but hourglass-prone formulation and an essentially non-hourglass formulation for elastic materials, employing a single limiter to dynamically adjust the weighting between the two formulations. Crucially, the dimensionless characteristic of the formulation enables seamless handling of complex material models. To validate the effectiveness of our formulation, we conduct simulations across a range of benchmark cases involving elastic, plastic, and anisotropic materials. To illustrate its versatility, we apply the formulation to simulate a complex scenario involving viscous plastic Oobleck material, contacts, and very large deformation. Our work addresses a critical gap in TLSPH simulations by offering a unified approach to mitigate hourglass modes, enhancing the reliability and accuracy of simulations across diverse material models and complex scenarios.

Essentially non-hourglass SPH elastic dynamics

More than two decades ago, the numerical instabilities of particle clustering and non-physical fractures encountered in the simulations of typical elastic dynamics problems using updated Lagrangian smoothed particle hydrodynamics (ULSPH) was identified as tensile instability. Despite continuous efforts in the past, a satisfactory resolution for the simulations of these problems has remained elusive. In this paper, the concept of hourglass modes, other than tensile instability, is first explored for the discretization of shear force, arguing that the former may actually lead to the numerical instabilities in these simulations. Based on such concept, we present an essentially non-hourglass formulation by utilizing the Laplacian operator which is widely used in fluid simulations. Together with the dual-criteria time stepping, adopted into the simulation of solids for the first time to significantly enhance computational efficiency, a comprehensive set of challenging benchmark cases is used to showcase that our method achieves accurate and stable SPH elastic dynamics.

Derivation and implementation of one-point quadrature quadrilateral shell element with MITC4+ method (MITC4+R)

In this work, a novel degenerated shell element called MITC4+R is developed, which is based on the MITC4+ (an improved MITC quadrilateral shell element) and reduced integration with one point quadrature. The assumed natural strain method is employed to construct the membrane and shear strains. A physical stabilization term is derivated after the assumed natural strains are decomposed into the linear and a constant term with respect to the natural coordinates. The hourglass modes owing to one point quadrature is controlled in the computation of deformation by introducing this physical stabilization term. The developed degenerated shell element is successfully implemented into the implicit static and explicit dynamic analyses. In order to verify its computational accuracy and efficiency, several standard benchmarks and practical problems are performed. These results show that the MITC4+R element eliminates various locking problems common to shell elements, and it significantly improves the computational efficiency of the original MITC4+ element. Meanwhile, the MITC4+R owns the same accuracy and highly consistent convergence as the original MITC4+ element, making it more promising for practical applications.

An essentially non-hourglass 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 essentially non-hourglass formulation based on volumetric-deviatoric 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 essentially suppress 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.

A new higher-order mixed four-node quadrilateral finite element for static bending analysis of functionally graded plates

In this paper, a new C0 four-node quadrilateral plate element (MiQ4) is proposed for the static bending analysis of functionally graded (FG) plates based on a C0 higher-order shear deformation theory. By introducing a quadratic shape function for higher-order terms and a constant shear result, which is eliminated at the element level, the transverse shear stresses are free on two surfaces of the plates. The present finite element model has high accurateness and a fast rate of convergence, especially for both thin and very thin plates. In addition, the proposed element is insensitive to distorted mesh, hence, it is compatible with the investigation of complex geometrical plates. Although the proposed element does not need any shear correction coefficients and selected or reduced integrations, the shear-locking phenomenon does not appear even for very thin plates. Therefore, the computation cost and time are reduced and the spurious deformation or hourglass modes are avoided. A large number of examples have been exhibited that help to validate the convergence rate, correctness, and reliability of the proposed element. Then the present element is applied for analysis of FG structures to demonstrate the potential applications of the present plate element. Significantly, the paper provides a good element that can be used in many areas of engineering for the design, testing, analysis, and optimization of complex structures to reduce the computation cost and time as well as increase the reliability of the numerical results.

Voxel‐based finite elements with hourglass control in fast Fourier transform‐based computational homogenization

AbstractThe power of fast Fourier transform (FFT)‐based methods in computational micromechanics critically depends on a seamless integration of discretization scheme and solution method. In contrast to solution methods, where options are available that are fast, robust and memory‐efficient at the same time, choosing the underlying discretization scheme still requires the user to make compromises. Discretizations with trigonometric polynomials suffer from spurious oscillations in the solution fields and lead to ill‐conditioned systems for complex porous materials, but come with rather accurate effective properties for finitely contrasted materials. The staggered grid discretization, a finite‐volume scheme, is devoid of bulk artifacts in the solution fields and works robustly for porous materials, but does not handle anisotropic materials in a natural way. Fully integrated finite‐element discretizations share the advantages of the staggered grid, but involve a higher memory footprint, require a higher computational effort due to the increased number of integration points and typically overestimate the effective properties. Most widely used is the rotated staggered grid discretization, which may also be viewed as an underintegrated trilinear finite element discretization, which does not impose restrictions on the constitutive law, has fewer artifacts than Fourier‐type discretizations and leads to rather accurate effective properties. However, this discretization comes with two downsides. For a start, checkerboard artifacts are still present. Second, convergence problems occur for complex porous microstructures. The work at hand introduces FFT‐based solution techniques for underintegrated trilinear finite elements with hourglass control. The latter approach permits to suppress local hourglass modes, which stabilizes the convergence behavior of the solvers for porous materials and removes the checkerboards from the local solution field. Moreover, the hourglass‐control parameter can be adjusted to “soften” the material response compared to fully integrated elements, using only a single integration point for nonlinear analyses at the same time. To be effective, the introduced technology requires a displacement‐based implementation. The article exposes an efficient way for doing so, providing minimal interfaces to the most commonly used solution techniques and the appropriate convergence criterion.

Quadrature rules for isogeometric shell formulations: Study using a real-world application about metal forming

This paper studies quadrature rules for simulating large deformations of shells using isogeometric analysis. Several recently proposed rules and their effects on a real-world application known as incremental sheet forming are investigated. It is observed that, when tackling real-world applications, unexpected problems arise and, therefore, theoretical studies only with manufactured solutions are not enough for a complete verification of a method. The chosen application reveals problems with certain quadratures and that some simple stabilization strategies cannot completely suppress hourglass modes. Additionally, the effects of quadrature rules on the total computational costs are demonstrated and the influence of the maximum stable time step is assessed using a highly demanding simulation.

Simple and Accurate Eight-Node and Six-Node Solid-Shell Elements with Explicit Element Stiffness Matrix Based on Quasi-Conforming Element Technique

Based on the quasi-conforming (QC) element technique, accurate and reliable eight-node and six-node solid-shell elements are presented in this paper. These QC solid-shell elements can alleviate shear and Poisson thickness locking by appropriately interpolating the strain fields over the element domain, and they are completely free from hourglass modes by ensuring the rank sufficiency of the element stiffness matrix a priori. Furthermore, the element stiffness matrices of the present elements are evaluated explicitly rather than resorting to the numerical integration, which leads to a high computational efficiency. The QC solid-shell elements with the properly interpolated element strain fields can rigorously pass both membrane and bending patch tests. The popular benchmark problems are used to evaluate the performance of the QC solid-shell elements. The numerical results show that the present QC solid-shell elements yield not only accurate displacements but also good stress results for all the stress components. Particularly, the present QC solid-shell elements are capable of giving quite accurate results even with very coarse mesh.

Numerical Considerations for the Implementation of a Comprehensive Composite Damage Model

A comprehensive continuum damage mechanics model [1] had been developed to capture the detailed behaviour of a composite structure under a crushing load. This paper explores some of the difficulties encountered in the implementation of this model and their mitigation. The use of reduced integration element and a strain softening model both negatively affect the accuracy and stability of the simulation. Damage localisation effects demanded an accurate measure of characteristic length. A robust algorithm for determining the characteristic length was implemented. Testing showed that this algorithm produced marked improvements over the use of the default characteristic length provided by Abaqus. Zero-energy or hourglass modes, in reduced integration elements, led to reduced resistance to bending. This was compounded by the strain softening model, which led to the formation of elements with little resistance to deformation that could invert if left unchecked. It was shown, through benchmark testing, that by deleting elements with excess distortions and controlling the mesh using inbuilt distortion/hourglass controls, these issues can be alleviated. These techniques contributed significantly to the viability and usability of the damage model.

Stability of 3D Textile Composite Reinforcement Simulations: Solutions to Spurious Transverse Modes

The simulation of thick 3D composite reinforcement forming brings to light new modeling challenges. The specific anisotropic material behavior due to the possible slippage between fibers induces, among other phenomena, the development of spurious transverse modes in bending-dominated 3D simulations. To obtain coherent finite element responses, two solutions are proposed. The first one uses a simple assumed strain formulation usually prescribed to prevent volumetric locking. This solution avoids spurious transverse modes by stiffening of the hourglass modes. Nevertheless the deformation obtained by this approach still suffers from the inability of the standard continuum mechanics of Cauchy to describe fibrous material deformation. The second proposed approach is based on the introduction of a bending stiffness which both avoids the spurious transverse modes and also improves the global behavior of the element formulation by enriching the underlying continuum. To emphasize the differences between different formulations, element stiffnesses are explicitly calculated and compared.

Developments of the mixed grid isogeometric Reissner–Mindlin shell: Serendipity basis and modified reduced quadrature

In this paper, we use the Serendipity basis in the isogeometric Reissner–Mindlin shell formulation to facilitate the fiber vector definition. The Serendipity basis and the NURBS basis are used here to express the shell fiber rotations and the mid-surface translations respectively. Results show that they perform nearly the same with the Lagrange/NURBS formulation, but with savings in the number of rotational degrees of freedom. We also study the transverse shear locking phenomenon and give a modified reduced quadrature scheme. It can improve the efficiency and to some extent relieve the locking without introducing hourglass modes. The linear decreasing, from outer elements to inner elements, quadrature point distribution also prevents the oscillation of the solutions in the coarse meshes.

Uniform Gradient Element Formulation with Hourglass Control Scheme for Solving Fully Coupled Finite-Element Equations for Saturated Soils

AbstractA uniform gradient (UG) element formulation for a four-node quadrilateral element is derived and implemented for solving the fully coupled governing equations for the dynamics of a saturated porous medium. An hourglass control scheme is also derived for eliminating the hourglass modes that arise when using the UG element formulation. The effectiveness of the new formulations is investigated by simulating the quasi-static and dynamics response of a footing and the dynamic response of an earth dam. The results show that the level of excitation of hourglass modes is problem-dependent and mesh-dependent. The proposed method for controlling the hourglass modes is shown to be effective with minimal effect on the computed response.

Improved numerical integration for locking treatment in isogeometric structural elements. Part II: Plates and shells

B-spline reduced quadrature rules are proposed in the context of isogeometric analysis. When performing a full Gaussian integration, the high regularity provided by spline basis functions strengthens the locking phenomena and deteriorates the performance of Reissner–Mindlin elements. The uni-dimensional B-spline-based quadrature rules, given in a previous paper (part I), are extended to multi-dimensional problems such as plates and shells. The improved reduced integration schemes are constructed using a tensor product of the uni-dimensional schemes. A single numerical quadrature is performed for bending, transverse shear and membrane terms, without introducing Hourglass modes. The proposed isogeometric reduced elements are free from membrane and transverse shear locking. Convergence is first assessed in plate problems with several aspect ratios and then in the shell obstacle course problems. The resulting under-integrated elements exhibit better accuracy and computational efficiency.

Microstructurally induced computational and material instabilities

A dislocation-density based finite strain crystal plasticity finite element model and an hourglass strain measure orthogonal to the linear velocity field were used to investigate microstructural effects on deformation and hourglass instability. An aluminum aggregate was modeled in quasi-static plane strain tension and compression, and the effects of the presence of manganese-based dispersed particles were investigated. Both local plastic deformation and the presence of these dispersed particles can trigger hourglass instabilities, which can be mistaken for deformation or failure modes. These hourglass modes occur due to the difference in the deformability of neighboring elements or regions and the associated energy modes associated with hourglassing, and the proposed computational methodology can be used to delineate between failure modes and numerical instabilities.

A new prismatic solid-shell element 'SHB6' : assumed-strain formulation and evaluation on benchmark problems

In this paper, the formulation of a new six-node solid–shell element denoted (SHB6) is proposed. This prismatic element is based on a purely three-dimensional approach, and hence has displacements as the only degrees of freedom. A reduced integration scheme is adopted consisting of one-point in-plane quadrature and an arbitrary number of integration points, with a minimum number of two, distributed along the ‘thickness’ direction. Moreover, in order to enhance its performance and to greatly reduce most locking effects, specific projections are introduced based on the assumed-strain method. The resulting derivation can then be used to model thin structural problems, while taking into account the various through-thickness phenomena. A careful analysis of potential stiffness matrix rank deficiencies reveals that no hourglass modes need to be controlled. However, without assumed-strain method, the element exhibits some shear and thickness-type locking, which is common in linear triangular elements associated with constant strain states. After the formulation of the element is detailed, its performance is assessed through a set of representative benchmark problems illustrating its capabilities in various situations. More specifically, this prismatic solid–shell element proves to be an essential complement to the SHB8PS hexahedral element in meshing arbitrarily complex geometries.

Isogeometric collocation for elastostatics and explicit dynamics

We extend the development of collocation methods within the framework of Isogeometric Analysis (IGA) to multi-patch NURBS configurations, various boundary and patch interface conditions, and explicit dynamic analysis. The methods developed are higher-order accurate, stable with no hourglass modes, and efficient in that they require a minimum number of quadrature evaluations. The combination of these attributes has not been obtained previously within standard finite element analysis.

Assumed-strain solid–shell formulation for the six-node finite element SHB6: evaluation on non-linear benchmark problems

Because accuracy and efficiency are the main features expected within the finite element (FE) method, the current contribution proposes a six-node prismatic solid–shell, denoted (SHB6). The formulation is extended here to geometric and material nonlinearities, and focus will be placed on its validation on nonlinear benchmark problems. This type of FE is specifically designed for the modeling of thin structures, by combining several useful shell features with some well-known solid element advantages. Therefore, the resulting derivation only involves displacement degrees of freedom as it is based on a fully 3D approach. Some of the motivation behind this formulation is to allow a natural mesh connection in problems where both structural (shell/plate) and continuum (solid) elements need to be simultaneously used. Another major interest of this prismatic solid–shell is to complement meshes that use hexahedral solid–shell FE, especially when free mesh generation tools are employed. To achieve an efficient formulation, the assumed-strain method is combined with an in-plane one-point quadrature scheme. These techniques are intended to reduce both locking phenomena and computational cost. A careful analysis of possible stiffness matrix rank deficiencies demonstrates that this reduced integration procedure does not induce hourglass modes and thus no stabilization is required.

Application of the continuum shell finite element SHB8PS to sheet forming simulation using an extended large strain anisotropic elastic–plastic formulation

This paper proposes an extension of the SHB8PS solid–shell finite element to large strain anisotropic elasto-plasticity, with application to several non-linear benchmark tests including sheet metal forming simulations. This hexahedral linear element has an arbitrary number of integration points distributed along a single line, defining the “thickness” direction; and to control the hourglass modes inherent to this reduced integration, a physical stabilization technique is used. In addition, the assumed strain method is adopted for the elimination of locking. The implementation of the element in Abaqus/Standard via the UEL user subroutine has been assessed through a variety of benchmark problems involving geometric non-linearities, anisotropic plasticity, large deformation and contact. Initially designed for the efficient simulation of elastic–plastic thin structures, the SHB8PS exhibits interesting potentialities for sheet metal forming applications—both in terms of efficiency and accuracy. The element shows good performance on the selected tests, including springback and earing predictions for Numisheet benchmark problems.

A Selective Reduced Integration 8-Node Hexahedral Element by Assumed Strain for Coining Simulation

A selective reduced integration 8-node hexahedral element for coining simulation is developed in this paper. The element is free of volume locking by assumed strain method. The standard velocity gradient matrix is derived in which the shear items are ignored to avoid shear locking. Hourglass modes are successfully suppressed without user-input parameters. The element is successfully employed in the coining simulation package - COINFORM. Numerical tests and experiments of a typical coin are carried out to show the good performances of the element.

The consistency of pressure‐gradient approximations used in multi‐dimensional shock hydrodynamics

AbstractThis paper presents a study of the consistency properties of the pressure‐gradient approximation used in multi‐dimensional finite‐element shock hydrodynamics codes today. In specific, consideration is given to the so‐called ‘bent‐element blues’ problem associated with the pressure‐gradient approximation when using the Q1Q0 element. On arbitrary grids comprised of distorted elements, the piecewise‐constant representation of the pressure field leads to a low‐order pressure‐gradient approximation at the global (nodal) level. This results in spurious nodal forces that are not aligned with the pressure gradient. There are several side‐effects of this behavior that include (a) incorrectly exciting physical modes in problems that exhibit unstable behavior, e.g. Rayleigh–Taylor problems (both magnetic and hydrodynamic), (b) potentially seeding hourglass modes, and (c) exhibiting non‐stationary behavior for steady‐state problems. A series of commonly used pressure‐gradient approximations are reviewed and evaluated based on linear consistency—the ability of the approximation to annihilate constant terms and exactly reproduce a linear gradient. The deeper theoretical issues associated with the proper selection of function spaces for the finite‐element hydro formulation are not discussed here. There are two gradient approximations that use piecewise‐constant data and deliver a consistent pressure‐gradient approximation on arbitrary grids. The first is the well‐known least‐squares gradient construction, and the second is a corrected gradient approximation that imposes linear consistency at the (global) nodal level. At the time of this writing, the corrected gradient approximation appears to be the most viable candidate for resolving the consistency issues associated with the Q1Q0 element technology. Copyright © 2009 John Wiley & Sons, Ltd.