Mass Lumping Research Articles

Efficient simulation of waves in heterogeneous domains using the scaled boundary finite element method

AbstractSimulation of wave propagation phenomena is very challenging due to the restrictions placed on the spatial as well as on the temporal discretization. Image‐based domain discretization using hierarchical meshes (2D polygonal elements) in conjunction with the scaled boundary finite element method provides an efficient numerical simulation tool. In this setting, the computational effort is reduced as only a limited number of unique cell patterns are formed. This is beneficial while computing element matrices in the preprocessing stage. When solving the dynamic system, the advantages of similar elements can only be fully exploited for the explicit time‐stepping schemes in connection with mass lumping when the domain is discretized using low‐order elements (p ≤ 2). On the other hand, the time step size may be limited by the stability criterion. Moreover, high‐order elements require fewer degrees of freedom. Hence for a specific application, it is not straightforward to predict superiority of one time‐stepping scheme over the other. A comparative study of implicit and explicit time‐stepping schemes is presented using an artificial two‐material model.

A nodal immersed finite element-finite difference method

The immersed finite element-finite difference (IFED) method is a computational approach to modeling interactions between a fluid and an immersed structure. The IFED method uses a finite element (FE) method to approximate the stresses, forces, and structural deformations on a structural mesh and a finite difference (FD) method to approximate the momentum and enforce the incompressibility of the entire fluid-structure system on a Cartesian grid. The fundamental approach used by this method follows the immersed boundary framework for modeling fluid-structure interaction (FSI), in which a force spreading operator prolongs structural forces to a Cartesian grid, and a velocity interpolation operator restricts a velocity field defined on that grid back onto the structural mesh. With an FE structural mechanics framework, force spreading first requires that the force itself be projected onto the finite element space. Similarly, velocity interpolation requires projecting velocity data onto the FE basis functions. Consequently, evaluating either coupling operator requires solving a matrix equation at every time step. Mass lumping, in which the projection matrices are replaced by diagonal approximations, has the potential to accelerate this method considerably. This paper provides both numerical and computational analyses of the effects of this replacement for evaluating the force projection and for the IFED coupling operators. Constructing the coupling operators also requires determining the locations on the structure mesh where the forces and velocities are sampled. Here we show that sampling the forces and velocities at the nodes of the structural mesh is equivalent to using lumped mass matrices in the IFED coupling operators. A key theoretical result of our analysis is that if both of these approaches are used together, the IFED method permits the use of lumped mass matrices derived from nodal quadrature rules for any standard interpolatory element. This is different from standard FE methods, which require specialized treatments to accommodate mass lumping with higher-order shape functions. Our theoretical results are confirmed by numerical benchmarks, including standard solid mechanics tests and examination of a dynamic model of a bioprosthetic heart valve.

A mixed selective edge-based smoothed PFEM with second-order cone programming for geotechnical large deformation analysis

Traditional particle finite element method (PFEM) with a six-node triangular (T6) element using an iteration algorithm often suffers from high computational costs and non-convergence problems. This paper proposes a novel dynamic implicit optimisation based smoothed PFEM for solving geotechnical large deformation problems. The governing equations are treated as an equivalent min–max optimisation that is recast as a second-order cone programming (SOCP) problem. Owing to the particularity of the min–max optimisation problem, which needs to simultaneously interpolate displacement and stress field, a novel mixed constant-stress selective edge-based smoothed strain element is proposed in which strain and stress are constant in the smoothing domain. The two-level mesh repartitioning scheme is implemented into the proposed method to overcome volumetric locking. Next, a mass lumping scheme using nodal integration is developed to improve computational efficiency. An efficient and simple variable mapping scheme is finally proposed and implemented into PFEM to tackle large deformation problems. Several classical statics and dynamics examples are used to validate the proposed method. All results demonstrate that the proposed method only using the lower-order element offers high accuracy, high efficiency, mesh insensitivity and is free of volumetric locking for geotechnical problems under both small and large deformation.

More Continuous Mass-Lumped Triangular Finite Elements

When solving the wave equation with finite elements, mass lumping allows for explicit time stepping, avoiding the cost of a lower-upper decomposition of the large sparse mass matrix. Mass lumping on the reference element amounts to numerical quadrature. The weights should be positive for stable time stepping and preserve numerical accuracy. The standard triangular polynomial elements, except for the linear element, do not have these properties. Accuracy can be preserved by augmenting them with higher-degree polynomials in the interior. This leaves the search for elements with positive weights, which were found up to degree 9 by various authors. The classic accuracy condition, however, is too restrictive. A sharper, less restrictive condition recently led to new mass-lumped tetrahedral elements up to degree 4. Compared to the known ones up to degree 3, they have less nodes and are computationally more efficient. The same criterion is applied here to the construction of triangular elements. For degrees 2 to 4, these turn out to be identical to the known ones. For degree 5, the number of nodes is the same as for the known element, but now there are infinitely many solutions. Some of these have a considerably larger stability limit for time stepping. For degree 6, two elements are found with less nodes than the known ones. For degree 7, one element with less nodes was found but with a negative weight, making it useless for time stepping with the wave equation. If the number of nodes is the same as for the classic element, there are now infinitely many solutions. Numerical tests for a homogeneous wave-propagation problem with a point source confirm the expected accuracy of the new elements. Some of them require less compute time than those obtained with the more restrictive accuracy criterion.

An implicit monolithic AFC stabilization method for the CG finite element discretization of the fully-ionized ideal multifluid electromagnetic plasma system

This study considers an implicit finite element formulation for an ideal fully-ionized multifluid electromagnetic plasma system. The formulation is based on fully-implicit Runge-Kutta time discretizations and a monolithic discrete algebraic flux corrected (AFC) continuous Galerkin (CG) spatial discretization of the coupled system. The AFC approach adds scalar artificial diffusion to the high-order, semi-discrete Galerkin method and uses mass lumping in the time derivative term. The result is a low-order method that attempts to enforce local-extremum-diminishing properties for the hyperbolic system. An element-based iterative limiter is applied to reduce the amount of artificial diffusion that is used in regions where the solution is smooth and the additional stabilization is not required. Two models are considered for the electromagnetics portion of the system: an electrostatic model, and a full Maxwell system with a parabolic divergence cleaning approach that enforces the required involutions on the electric and magnetic fields. Results are presented that demonstrate the accuracy and robustness of the formulation for smooth and discontinuous solutions to challenging plasma physics problems. This includes a demonstration that the solution of the full multifluid system yields the expected behavior in the ideal shock-MHD limit.

High Order Mixed Finite Elements with Mass Lumping for Elasticity on Triangular Grids

A family of conforming mixed finite elements with mass lumping on triangular grids are presented for linear elasticity. The stress field is approximated by symmetric $H({\rm div})-P_k (k\geq 3)$ polynomial tensors enriched with higher order bubbles so as to allow mass lumping, which can be viewed as the Hu-Zhang elements enriched with higher order interior bubble functions. The displacement field is approximated by $C^{-1}-P_{k-1}$ polynomial vectors enriched with higher order terms to ensure the stability condition. For both the proposed mixed elements and their mass lumping schemes, optimal error estimates are derived for the stress with $H(\rm div)$ norm and the displacement with $L^2$ norm. Numerical results confirm the theoretical analysis.

A Nodal Immersed Finite Element-Finite Difference Method

The immersed finite element-finite difference (IFED) method is a computational approach to modeling interactions between a fluid and an immersed structure. This method uses a finite element (FE) method to approximate the stresses and forces on a structural mesh and a finite difference (FD) method to approximate the momentum of the entire fluid-structure system on a Cartesian grid. The fundamental approach used by this method follows the immersed boundary framework for modeling fluid-structure interaction (FSI), in which a force spreading operator prolongs structural forces to a Cartesian grid, and a velocity interpolation operator restricts a velocity field defined on that grid back onto the structural mesh. Force spreading and velocity interpolation both require projecting data onto the finite element space. Consequently, evaluating either coupling operator requires solving a matrix equation at every time step. Mass lumping, in which the projection matrices are replaced by diagonal approximations, has the potential to accelerate this method considerably. Constructing the coupling operators also requires determining the locations on the structure mesh where the forces and velocities are sampled. Here we show that sampling the forces and velocities at the nodes of the structural mesh is equivalent to using lumped mass matrices in the coupling operators. A key theoretical result of our analysis is that if both of these approaches are used together, the IFED method permits the use of lumped mass matrices derived from nodal quadrature rules for any standard interpolatory element. This is different from standard FE methods, which require specialized treatments to accommodate mass lumping with higher-order shape functions. Our theoretical results are confirmed by numerical benchmarks, including standard solid mechanics tests and examination of a dynamic model of a bioprosthetic heart valve.

FREE VIBRATION OF SKEW LAMINATES – A BRIEF REVIEW AND SOME BENCHMARK RESULTS

This study investigates and reviews prior research works on skew composite laminates. The equivalent single layer theories are explored and discussed. An exhaustive review on static and dynamic analysis of composite skew laminates is also presented. Subsequently, a nine node isoparametric plate bending element is used for free vibration analysis of laminated composite skew plate with central skew cut out. The effect of shear deformation is incorporated in the formulation considering first order shear deformation theory. Two types of mass lumping schemes are analysed to study the effect of rotary inertia. Certain numerical examples of plates having different skew angles, skew cut out sizes, boundary conditions, thickness ratios (h/a), aspect ratios (a/b), fiber orientations and number of layers are solved which will be useful for benchmarking of future studies.

DYNAMIC ANALYSIS OF PLATES WITH CUT-OUT CARRYING CONCENTRATED AND DISTRIBUTED MASS

An isoparametric plate bending element with nine nodes is used in this paper for dynamic analysis of isotropic cut-out plate having concentrated and uniformly distributed mass on the plate. The Mindlin’s first-order shear deformation theory (FSDT) is used in the present finite element formulation. Two proportionate mass lumping schemes are used. The effect of rotary inertia is included in one of the mass lumping schemes in the present element formulation. Dynamic analysis of rectangular isotropic plates with cut-out having different side ratio, thickness ratio and boundary condition is analysed using a finite element method. The present results are compared with the published results. Some new results on isotropic plates with cut-out having different side ratio, ratio of side-to-thickness of the plate, different position and size of cut-out in plates subjected to transversely concentrated and distributed mass are presented.

Fast mass lumped multiscale wave propagation modelling

Abstract In this paper we investigate the use of a mass lumped fully explicit time-stepping scheme for the discretization of the wave equation with underlying material parameters that vary at arbitrarily fine scales. We combine the leapfrog scheme for the temporal discretization with the multiscale technique known as localized orthogonal decomposition (LOD) for the spatial discretization. To speed up the method and to make it fully explicit, a special mass lumping approach is introduced that relies on an appropriate interpolation operator. This operator is also employed in the construction of the LOD and is a key feature of the approach. We prove that the method converges with second order in the energy norm, with a leading constant that does not depend on the scales at which the material parameters vary. We also illustrate the performance of the mass lumped method in a set of numerical experiments.

High order transition elements: The xNy-element concept, Part II: Dynamics

In this article, the construction of versatile transition elements for applications in dynamics is comprehensively discussed. Based on the transfinite mapping technique, also known as Coons–Gordon interpolation, piece-wise shape functions are derived such that an arbitrary number of elements with different ansatz spaces is coupled conformingly. This is an important prerequisite to enable the use of local mesh refinement strategies for regular quadrilateral discretizations. Due to the special construction of the shape functions, there are no hanging nodes that need to be taken care of. The focus in this contribution is on dynamics, and therefore, Lagrange polynomials are taken as the basis to construct shape functions for arbitrary transition elements. As a consequence, the Kronecker-delta and partition of unity properties are recovered. These properties are crucial to diagonalize the mass matrix using standard mass lumping techniques such as row-summing and diagonal scaling. The performance of the proposed family of elements is assessed by means of several benchmark examples, where the numerical rates of convergence are determined for both consistent and lumped mass matrix formulations.

Scale-bridging with the extended/generalized finite element method for linear elastodynamics

This paper presents an extended/generalized finite element method for bridging scales in linear elastodynamics in the absence of scale separation. More precisely, the GFEMgl framework is expanded to enable the numerical solution of multiscale problems through the automated construction of specially-tailored shape functions, thereby enabling high-fidelity finite element modeling on simple, fixed finite element meshes. This introduces time-dependencies in the shape functions in that they are subject to continuous adaptation with time. The temporal aspects of the formulation are investigated by considering the Newmark- $$\beta $$ time integration scheme, and the efficacy of mass lumping strategies is explored in an explicit time-stepping scheme. This method is demonstrated on representative wave propagation examples as well as a dynamic fracture problem to assess its accuracy and flexibility.

Explicit edge-based smoothed numerical manifold method for transient dynamic modeling of two-dimensional stationary cracks

Modeling and prediction of dynamic behaviors of cracked bodies are vital to the safety evaluation of structures. The numerical manifold method (NMM) has achieved considerable success in crack analysis as it provides a powerful representation of complex discontinuities. Strain smoothing has been incorporated into enriched methods such as NMM to improve their accuracy, efficiency, and tolerance to mesh distortion. However, conventional strain smoothing cannot reproduce the large spatial variance in the strain surrounding crack tips. Consequently, the accuracy is reduced when it is applied to crack-tip enrichment bases. High-fidelity representation of crack-tip fields can be achieved by crack-tip enrichment. Therefore, it is desirable to combine compatible strain for the enriched areas with edge-based smoothed strain for the remaining areas. When explicit time integration is employed, it is beneficial to use a lumped mass matrix because it provides a larger critical time step and a straightforward solution of the linear algebraic equation. A local-approximation-based mass lumping strategy is used to obtain a block-diagonal mass matrix with small blocks. Finally, the proposed methodology is applied to various static and dynamic crack analyses, and highly accurate and stable results are obtained.

A massively parallel explicit solver for elasto-dynamic problems exploiting octree meshes

Typical areas of application of explicit dynamics are impact, crash test, and most importantly, wave propagation simulations. Due to the numerically highly demanding nature of these problems, efficient automatic mesh generators and transient solvers are required. To this end, a parallel explicit solver exploiting the advantages of balanced octree meshes is introduced. To avoid the hanging nodes problem encountered in standard finite element analysis (FEA), the scaled boundary finite element method (SBFEM) is deployed as a spatial discretization scheme. Consequently, arbitrarily shaped star-convex polyhedral elements are straightforwardly generated. Considering the scaling and transformation of octree cells, the stiffness and mass matrices of a limited number of unique cell patterns are pre-computed. A recently proposed mass lumping technique is extended to 3D yielding a well-conditioned diagonal mass matrix. This enables us to leverage the advantages of explicit time integrator, i.e., it is possible to efficiently compute the nodal displacements without the need for solving a system of linear equations. We implement the proposed scheme together with a central difference method (CDM) in a distributed computing environment. The performance of our parallel explicit solver is evaluated by means of several numerical benchmark examples, including complex geometries and various practical applications. A significant speedup is observed for these examples with up to one billion of degrees of freedom and running on up to 16,384 computing cores.

Analysis of a fully discrete approximation for the classical Keller–Segel model: Lower and a priori bounds

This paper is devoted to constructing approximate solutions for the classical Keller–Segel model governing chemotaxis. It consists of a system of nonlinear parabolic equations, where the unknowns are the average density of cells (or organisms), which is a conserved variable, and the average density of chemoattractant.The numerical proposal is made up of a crude finite element method together with a mass lumping technique and a semi-implicit Euler time integration. The resulting scheme turns out to be linear and decouples the computation of variables. The approximate solutions keep lower bounds – positivity for the cell density and nonnegativity for the chemoattractant density–, are bounded in the L1(Ω)-norm, satisfy a discrete energy law, and have a priori energy estimates. The latter is achieved by means of a discrete Moser–Trudinger inequality. As far as we know, our numerical method is the first one that can be encountered in the literature dealing with all of the previously mentioned properties at the same time. Furthermore, some numerical examples are carried out to support and complement the theoretical results.

A Mid-Node Mass Lumping Scheme for Accurate Structural Vibration Analysis with Serendipity Finite Elements

A mid-node mass lumping scheme is proposed to formulate the lumped mass matrices of serendipity elements for accurate structural vibration analysis. Since the row-sum technique leads to unacceptable negative lumped mass components for serendipity elements, the diagonal scaling HRZ method is frequently employed to construct lumped mass matrices of serendipity elements. In this work, through introducing a lumped mass matrix template that includes the HRZ lumped mass matrix as a special case, an analytical frequency accuracy measure is rationally derived with particular reference to the classical eight-node serendipity element. The theoretical results clearly reveal that the standard HRZ mass matrix actually does not offer the optimal frequency accuracy in accordance with the given lumped mass matrix template. On the other hand, by employing the nature of non-negative shape functions associated with the mid-nodes of serendipity elements, a mid-node lumped mass matrix (MNLM) formulation is introduced for the mass lumping of serendipity elements without corner nodal mass components, which essentially corresponds to the optimal frequency accuracy in the context of the given lumped mass matrix template. Both theoretical and numerical results demonstrate that MNLM yields better frequency accuracy than the standard HRZ lumped mass matrix formulation for structural vibration analysis.

A Second-Order Finite Element Method with Mass Lumping for Maxwell's Equations on Tetrahedra

We consider the numerical approximation of Maxwell's equations in the time domain by a second-order $H({curl})$ conforming finite element approximation. In order to enable the efficient application...

Effect of Computational Schemes on Coupled Flow and Geo-Mechanical Modeling of CO2 Leakage through a Compromised Well

Carbon capture, utilization, and storage (CCUS) describes a set of technically viable processes to separate carbon dioxide (CO2) from industrial byproduct streams and inject it into deep geologic formations for long-term storage. Legacy wells located within the spatial domain of new injection and production activities represent potential pathways for fluids (i.e., CO2 and aqueous phase) to leak through compromised components (e.g., through fractures or micro-annulus pathways). The finite element (FE) method is a well-established numerical approach to simulate the coupling between multi-phase fluid flow and solid phase deformation interactions that occur in a compromised well system. We assumed the spatial domain consists of a three-phases system: a solid, liquid, and gas phase. For flow in the two fluids phases, we considered two sets of primary variables: the first considering capillary pressure and gas pressure (PP) scheme, and the second considering liquid pressure and gas saturation (PS) scheme. Fluid phases were coupled with the solid phase using the full coupling (i.e., monolithic coupling) and iterative coupling (i.e., sequential coupling) approaches. The challenge of achieving numerical stability in the coupled formulation in heterogeneous media was addressed using the mass lumping and the upwinding techniques. Numerical results were compared with three benchmark problems to assess the performance of coupled FE solutions: 1D Terzaghi’s consolidation, Liakopoulos experiments, and the Kueper and Frind experiments. We found good agreement between our results and the three benchmark problems. For the Kueper and Frind test, the PP scheme successfully captured the observed experimental response of the non-aqueous phase infiltration, in contrast to the PS scheme. These exercises demonstrate the importance of fluid phase primary variable selection for heterogeneous porous media. We then applied the developed model to the hypothetical case of leakage along a compromised well representing a heterogeneous media. Considering the mass lumping and the upwinding techniques, both the monotonic and the sequential coupling provided identical results, but mass lumping was needed to avoid numerical instabilities in the sequential coupling. Additionally, in the monolithic coupling, the magnitude of primary variables in the coupled solution without mass lumping and the upwinding is higher, which is essential for the risk-based analyses.

Dynamic Analysis of Plates with Cut-Out Carrying Concentrated and Distributed Mass

An isoparametric plate bending element with nine nodes is used in this paper for dynamic analysis of isotropic cut-out plate having concentrated and uniformly distributed mass on the plate. The Mindlin’s first-order shear deformation theory (FSDT) is used in the present finite element formulation. Two proportionate mass lumping schemes are used. The effect of rotary inertia is included in one of the mass lumping schemes in the present element formulation. Dynamic analysis of rectangular isotropic plates with cut-out having different side ratio, thickness ratio and boundary condition is analysed using a finite element method. The present results are compared with the published results. Some new results on isotropic plates with cut-out having different side ratio, ratio of side-to-thickness of the plate, different position and size of cut-out in plates subjected to transversely concentrated and distributed mass are presented.

Modeling a Mechanical Molecular Spring Isolator with High-Static-Low-Dynamic-Stiffness Properties

A mechanical molecular spring isolator (MMSI) is proposed for the purpose of isolating the low-frequency vibration of a heavy payload. The MMSI is a passive vibration isolation technique mimicking molecular spring isolator characteristics of high-static-low-dynamic stiffness (HSLDS). An MMSI consists of a piston-cylinder container filled with the liquid and some hydraulic spring accumulators. The piston would support a lump of mass and be subjected to a specific external vibration excitation force. Those accumulators can get intercommunication by the liquid to produce the transformation from high static stiffness to low dynamic stiffness. The stiffness model of the MMSI with several identical accumulators is established based on the hydrostatic law. After that, some parameters that significantly influence the stiffness characteristics are studied. Results show that the stiffness property of this kind of MMSI demonstrates a piecewise linearity of three segments. It applies the averaging method to acquire amplitude-frequency and phase-frequency relationships of the piecewise linear vibration isolation system. An inevitable jump phenomenon may occur when the exciting force reaches the critical value. The vibration isolation performance is evaluated by energy transmissibility. Finally, an experimental prototype was designed to carry out quasi-static and dynamic experiments to verify the stiffness model and the dynamic properties as an HSLDS vibration isolator.