Implicit Time Integration Research Articles

A plane‐strain hydraulic fracture driven by a shear‐thinning Carreau fluid

AbstractWe study the propagation of a plane‐strain hydraulic fracture driven by a shear thinning fluid following a Carreau rheology. We restrict to the impermeable medium case and quantify in details the impact on fracture growth of the shear‐thinning properties of the fluid between the low and high shear‐rates Newtonian limits. We derive several dimensionless numbers governing the evolution of the solution. The propagation notably depends on the ratio between the two limiting viscosities, the fluid shear‐thinning index, a dimensionless fracture toughness and a characteristic time‐scale capturing the instant at which the fluid inside the fracture reaches the low‐shear rate Newtonian plateau. We solve the problem numerically using Gauss‐Chebyshev methods for the spatial discretization of the coupled hydro‐mechanical problem and a fully implicit time integration scheme. The solution evolves from an early time self‐similar solution equals to the Newtonian one for the large‐shear rate viscosity to a late time self‐similar solution equals to the low‐shear rate Newtonian solution. The transition period (corresponding to the shear thinning part of the rheology) exhibits features similar to the power law rheology, albeit quantitatively different. Comparisons of hydraulic fracture growth predictions obtained with a power‐law model confirm its inadequacy for realistic fluids used in practice compared to the more physical Carreau rheology: the Newtonian plateau at high and low shear rates cannot be neglected.

3D mixed virtual element formulation for dynamic elasto-plastic analysis

The virtual element method (VEM) for dynamic analyses of nonlinear elasto-plastic problems undergoing large deformations is outlined within this work. VEM has been applied to various problems in engineering, considering elasto-plasticity, multiphysics, damage, elastodynamics, contact- and fracture mechanics. This work focuses on the extension of VEM formulations towards dynamic elasto-plastic applications. Hereby low-order ansatz functions are employed in three dimensions with elements having arbitrary convex or concave polygonal shapes. The formulations presented in this study are based on minimization of potential function for both the static as well as the dynamic behavior. Additionally, to overcome the volumetric locking phenomena due to elastic and plastic incompressibility conditions, a mixed formulation based on a Hu-Washizu functional is adopted. For the implicit time integration scheme, Newmark method is used. To show the model performance, various numerical examples in 3D are presented.

Iterative Approximation of Preconditioning Matrices Through Krylov-Type Solver Iterations

Linear equation solvers require considerable computational time in many computer simulation methods, such as the stress analysis using the finite element method (FEM), or the fluid dynamics using an implicit time integration scheme. Because of their favorable nature to modern supercomputers, Krylov-type linear equation solvers have become dominant. Krylov-type solvers are usually used with preconditioners, which accelerate the convergence of Krylov solvers, and therefore developing robust preconditioners draws attention from researchers. The basic idea of preconditioning is to transform the original system to a system which can be solved more easily by using a matrix which resembles the original system but the associated linear system can be solved without any difficulties. In this study, we propose a new methodology of constructing such a matrix by updating the matrix using the information obtained through the Krylov solver computation. More precisely, we employ the limited memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) Hessian matrix update scheme. The proposed scheme is combined with the orthogonal minimum (ORTHOMIN) residual method, which accepts a variable preconditioner. As a result of performance tests, in the best case scenario, (1) with the Matrix Market matrices, our new method combined with Algebraic Multigrid (AMG) shows the successful convergence in 12 cases out of 13 problems, whilst the conventional AMG converged only in four cases and (2) with FEM problems, we obtained [Formula: see text] acceleration in the rate of converge and [Formula: see text] in computational time over the AMG preconditioned conjugate gradient (CG) solver. With those results, our newly developed algorithm provides robustness to the conventional preconditioning, whilst the computational speed is superior to conventional ones.

A contact dynamics code implementation for the simulation of asteroid evolution and regolith in the asteroid environment

Over the last decades, simulations by discrete elements methods (DEM) have proven to be a reliable analysis tool in various domains of science and engineering. By providing access to the local physical mechanisms, DEM allows the exploration of microscopic based phenomena related to particles properties and interactions in various conditions and to revisit constitutive equations consequently. The growing computer power and memory now allow us to handle large collections of grains of various shapes and sizes by DEM simulations and in particular, it offers new perspectives in the exploration of the behavior of asteroids seen as self-gravitating and cohesive granular aggregates. In this paper we describe the Contact Dynamics (CD) method, a class of DEM based on non-smooth mechanics, and its implementation in the open-source software LMGC90. In contrast to more classical approach, Hard- and Soft-Sphere DEM, the CD method is based on an implicit time integration of the equations of motion and on a non-regularized formulation of mutual exclusion between particles. This numerical strategy is particularly relevant to the study of dense granular assemblies (even of large size) because it does not introduce numerical artifacts due to contact stiffness. So that it can be used for Small Body research, we implement a parallelized kd-tree and monitor the performance of the code as it simulates a number of granular systems. We provide examples of the simulation of the accretion of self-gravitating aggregates as well as their rotational disruption. We use the routines at our disposal in the code to monitor their behavior through the entire evolution and find agreement with previous research.

Proper Generalized Decomposition with time adaptive space separation for transient wave propagation problems in separable domains

Transient wave propagation problems may involve rich discretizations, both in space and in time, leading to computationally expensive simulations, even for simple spatial domains. The Proper Generalized Decomposition (PGD) is an attractive model order reduction technique to address this issue, especially when the spatial domain is separable. In this work, we propose a space separation with a time adaptive number of modes to efficiently capture transient wave propagation in separable domains. We combine standard time integration schemes with this original space separated representation for empowering standard procedures. The numerical behavior of the proposed method is explored through several 2D wave propagation problems involving radial waves, propagation on long time analyses, and wave conversions. We show that the PGD solution approximates its standard finite element solution counterpart with acceptable accuracy, while reducing the storage needs and the computation time (CPU time). Numerical results show that the CPU time per time step linearly increases when refining the mesh, even with implicit time integration schemes, which is not the case with standard procedures.

On the development, verification, and validation of a discontinuous Galerkin solver for the Navier–Stokes equations

We present maDG, a newly developed code designed for the efficient parallel implicit solution of the 3D unstructured nodal discontinuous Galerkin discretization of the unsteady compressible Navier–Stokes equations. The code is being developed to provide an efficient, modular, and maintainable software platform for studying algorithmic and modeling issues arising in high-resolution CFD. In this paper we address some issues pertinent to the software architecture and testing implementation used towards the goal of implicit time integration of discontinuous Galerkin (DG) approximations.This effort is largely inspired by the life work of Antony Jameson and by his legendary FLO codes, which for more than four decades have provided a unique platform for the development of CFD methods, and enabled path-breaking advances in aerodynamics. We dedicate this paper to Antony on the occasion of his 85th birthday, with the utmost gratitude for his teaching and his friendship.

Investigation of Air Pocket Behavior in Pipelines Using Rigid Column Model and Contributions of Time Integration Schemes

This paper studies the air pressurization problem caused by a partially pressurized transient flow in a reservoir-pipe system. The purpose of this study is to analyze the performance of the rigid column model in predicting the attenuation of the air pressure distribution. In this regard, an analytic formula for the amplitude and frequency will be derived, in which the influential parameters, particularly, the driving pressure and the air and water lengths, on the damping can be seen. The direct effect of the driving pressure and inverse effect of the product of the air and water lengths on the damping will be numerically examined. In addition, these numerical observations will be examined by solving different test cases and by comparing to available experimental data to show that the rigid column model is able to predict the damping. However, due to simplified assumptions associated with the rigid column model, the energy dissipation, as well as the damping, is underestimated. In this regard, using the backward Euler implicit time integration scheme, instead of the classical fourth order explicit Runge–Kutta scheme, will be proposed so that the numerical dissipation of the backward Euler implicit scheme represents the physical dissipation. In addition, a formula will be derived to calculate the appropriate time step size, by which the dissipation of the heat transfer can be compensated.

A Novel Family of Two-Stage Implicit Time Integration Schemes for Structural Dynamics

In this paper, a new family of composite implicit time schemes is developed to overcome some shortcomings of the existing composite schemes. In the development, unconventional time approximations are used for the displacement and velocity vectors. The algorithmic parameters of the approximations are optimized to have improved numerical characteristics. This study also explains some difficulties of the existing implicit composite schemes in dealing with some special types of external force vectors when the splitting ratio is varied over the range of current time interval. Some drawbacks of the existing composite schemes due to the adjustable splitting ratio are resolved in the newly proposed family through different algorithmic parameters. The new schemes can handle the externally applied forces as simple as the Newmark method does, which is impossible in the existing composite schemes. Impact and excitation of elastic bar problems are numerically solved by using the new and existing two-stage schemes, and numerical results are analyzed, illustrating the merit of the proposed schemes.

Quantum Fokker-Planck modeling of degenerate electrons

An implicit and conservative numerical scheme is proposed for the isotropic quantum Fokker-Planck equation describing the evolution of degenerate electrons subject to elastic collisions with other electrons and ions. The electron-ion and electron-electron collision operators are discretized using a discontinuous Galerkin method, and the electron energy distribution is updated by an implicit time integration method. The numerical scheme is designed to satisfy all conservation laws exactly. Numerical tests and comparisons with other modeling approaches are shown to demonstrate the accuracy and conservation properties of the proposed method.

Insight to the Newmark Implicit Time Integration Method for Solving the Wave Propagation Problems

Insight to the Newmark Implicit Time Integration Method for Solving the Wave Propagation Problems

Implicit large Eddy simulation of the NASA CRM high-lift configuration near stall

Recent progress in adaptive high-order numerical methods, high-order mesh generation, and scalable implicit time integration approaches has made large eddy simulation (LES) more affordable than ever before. In the present study, we demonstrate the progress by performing implicit LES of a benchmark problem from the 3rd AIAA High-Lift Prediction Workshop: NASA's Common Research Model (CRM) high-lift configuration. The LES tool, hpMusic, is based on the flux reconstruction method capable of handling high-order mixed unstructured meshes. P-refinement studies are used to assess the mesh-dependence of the numerical solutions. Comparisons are also made with available experimental data, and a very good agreement was obtained in lift and drag coefficients between the present computation and wall-corrected experimental data.

On Hydromechanical Interaction during Propagation of Localized Damage in Rocks

This paper provides a mathematical description of hydromechanical coupling associated with propagation of localized damage. The framework incorporates an embedded discontinuity approach and addresses the assessment of both hydraulic and mechanical properties in the region intercepted by a fracture. Within this approach, an internal length scale parameter is explicitly employed in the definition of equivalent permeability as well as the tangential stiffness operators. The effect of the progressive evolution of damage on the hydro-mechanical coupling is examined and an evolution law is derived governing the variation of equivalent permeability with the continuing deformation. The framework is verified by a numerical study involving 3D simulation of an axial splitting test carried out on a saturated sample under displacement and fluid pressure-controlled conditions. The finite element analysis incorporates the Polynomial-Pressure-Projection (PPP) stabilization technique and a fully implicit time integration scheme.

Viscous and hyperviscous filtering for direct and large-eddy simulation

This work is dedicated to the solution filtering technique for performing direct and large-eddy simulation. It is shown that this approach is equivalent to the use of spectral viscosity as a possible ersatz of subgrid-scale modelling. In the framework of finite-difference schemes, the filter operator can be designed to ensure time consistency while easily controlling the level and scale selectivity of the dissipation thus introduced. Then, a new family of filter schemes is developed in order to represent both the molecular and artificial dissipations. The resulting viscous filter operator is straightforward to implement through a simple modification of its coefficients that depend on the molecular/artificial viscosity and the time step. A definitive advantage in terms of computational efficiency is obtained for computational configurations where the time step is restricted by the Fourier condition, as a simple alternative to implicit time integration of the viscous term. Numerical tests clearly show that this viscous filtering method is flexible, accurate and numerically stable at large Fourier number.

Time Integration Algorithms for Elasto-Viscoplastic Models with Multiple Hardening Laws for Geomaterials: Enhancement and Comparative Study

To describe the behaviours of geomaterials such as time-dependency, anisotropy and destructuration, multiple hardening parameters and laws are generally needed for application in advanced elasto-viscoplastic models. Time integration with stress updating is a key step in the application of elasto-viscoplastic models to engineering practice. However, the robustness of time integration algorithms for such complicated models has rarely been studied, creating difficulties in selecting and improving algorithms. This paper focuses on use of three typical implicit time integration algorithms—Katona, Stolle and cutting plane—for integration of an advanced elasto-viscoplastic model. First, all selected algorithms are improved to fit the characteristics of the advanced model with multiple hardening parameters and are combined with adaptive substepping procedures to enhance their performance. Then a step-changed undrained triaxial test is simulated at the integration point level, on the basis of which variations in iteration number and relative error of stresses with step size are investigated and compared. Furthermore, the advanced model using different algorithms is implemented into finite element code, with global convergence and calculation time investigated and compared for two boundary value problems: a biaxial test and an embankment. All comparisons demonstrate that the modified cutting plane algorithm with substepping is the most robust and efficient one, followed by the modified Stolle with substepping and the modified Katona with substepping, for an advanced model with multiple hardening parameters.

Survey of cubic B-spline implicit time integration methodin computational wave propagation

Recently an implicit time integration method based-on Cubic B-spline has been presented for solving the problems in structural dynamics. This method benefits from high accuracy and desire stability. In this paper, the presented method is developed for analyzing the wave propagation problems and the results are compared to the methods in the literature including a series of Bathe family methods (Noh-Bathe, Standard Bathe p1/B2-Bathe and pq-Bathe methods) and Newmark trapezoidal rule method. A numerical dispersion analysis is presented to evaluate the method in one and two wave propagation problems. Results indicate that the method under study has adequate performance in wave propagation as good as structural dynamic problems.

Nonlinear dynamic buckling analysis of imperfect viscoelastic composite laminated plates

The aim of this paper is to study the effects of linear viscoelastic behavior on dynamic buckling response of imperfect composite laminated plates subjected to in-plane dynamic loads by implementing semi-analytical finite strip method (FSM). The semi-analytical FSM converges with a comparatively small number of strips and correspondingly small number of degrees of freedom. Thus, it is usually implemented more easily and faster than many other computational methods. The governing equations are derived by using classical laminated plate theory (CLPT) and the behavior of plate is assumed to be geometrically nonlinear through Von-Karman assumptions. The Newmark's implicit time integration method in conjunction with the Newton-Raphson iteration are employed to solve the nonlinear governing equation. A Kelvin-Voigt viscoelastic model is considered, and the effects of viscosity coefficient, thickness of the layers of the composite plate and boundary conditions on the nonlinear dynamic buckling response are discussed. In order to justify the accuracy of the results, some of them are verified against those available in other sources. It is also shown that the nonlinear dynamic buckling response of an imperfect viscoelastic composite laminated plate is significantly different from the elastic one by considering different viscosity coefficients.

A ghost-cell immersed boundary method for unified simulations of flow over finite- and zero-thickness moving bodies at large CFL numbers

A ghost-cell immersed boundary method for unified simulations of flow over finite- and zero-thickness moving bodies at large Courant-Friedrichs-Lewy (CFL) numbers is presented. In order to handle such bodies in a unified manner, algorithms for interface construction and cell demarcation are proposed. The main challenge in treating zero-thickness bodies is to maintain sharpness and accuracy even at large CFL numbers with diminished spurious force oscillations. Thus, the effect of large CFL numbers on the solution accuracy of fluid-structure interaction (FSI) problems involving zero-thickness bodies is investigated and necessary treatments to preserve solution accuracy even at large CFL numbers are suggested. The present study suggests two treatments which are important in preserving the accuracy and stability of the solution: backward time integration for computational cells called ‘swept-cells’ and pressure boundary condition with mass conservation. Composite implicit time integration for the dynamic equation of a thin elastic structure is employed for a stable simulation of FSI at large CFL numbers. By using large time step sizes, the present method not only enhances computational efficiency, but also suppresses spurious force oscillations while maintaining the sharpness of an infinitesimally thin body. The efficacy and accuracy of the present method are examined through numerical examples.

3D Virtual Elements for Elastodynamic Problems

AbstractA virtual element framework for nonlinear elastodynamics is outlined within this work. The virtual element method (VEM) can be considered as an extension of the classical finite element method. While the finite element method (FEM) is restricted to the usage of regular shaped elements, VEM allows to use non‐convex shaped elements for the spatial discretization [1]. It has been applied to various engineering problems in elasticity and other areas, such as plasticity or fracture mechanics as outlined in [3, 4]. This work deals with the extension of VEM to dynamic problems. Low‐order ansatz functions in two and three dimensions, with elements being arbitrary shaped, are used in this contribution. The formulations considered in this framework are based on minimization of energy, where a pseudo potential is used for the dynamic behavior. While the stiffness‐matrix needs a suitable stabilization, the mass‐matrix can be calculated fully through the projection part. For the implicit time integration, Newmark‐Method is used. To show the performance of the method, various numerical examples in 2D and 3D are presented.

The influence of compatibility equations of non‐linear elasticity on convergence for spectral method solutions

AbstractIn structural mechanics the stresses are of a primary interest. Due to the required regularity of the approximation it is often inconvenient to obtain the stresses from the displacement methods. Mixed and hybrid methods are thus proposed to avoid this obstacle. The compatibility conditions should be fulfilled in order to obtain a unique solution.In this paper the compatibility equations of non‐linear elasticity are considered. Despite its very long history, this issue has not been satisfactorily resolved for non‐simply‐connected bodies until recently [1]. Its well known however, that the compatibility conditions are linearly dependent.The aim of the present contribution is to include the spectral methods [2] in order to improve the finite element solution procedure for the reversible elastodynamics problem [3], formulated for a large strain deformation with the deformation gradient tensor as one of the basic unknowns. In the case of the small strains the components of the deformation gradient tensor are of different magnitude. The main contribution is the derivation of the solution strategy in order to take this information into account. Applying this strategy to the implicit time integration method the exponent decrease of the condition numbers of the iteration matrices was achieved. Consequently the accuracy of the calculations should increase and the convergence for spectral method solutions should considerably improve. The theoretical findings are confirmed by a selected numerical example from structural mechanics.

Composite Backward Differentiation Formula for the Bidomain Equations.

The bidomain equations have been widely used to model the electrical activity of cardiac tissue. While it is well-known that implicit methods have much better stability than explicit methods, implicit methods usually require the solution of a very large nonlinear system of equations at each timestep which is computationally prohibitive. In this work, we present two fully implicit time integration methods for the bidomain equations: the backward Euler method and a second-order one-step two-stage composite backward differentiation formula (CBDF2) which is an L-stable time integration method. Using the backward Euler method as fundamental building blocks, the CBDF2 scheme is easily implementable. After solving the nonlinear system resulting from application of the above two fully implicit schemes by a nonlinear elimination method, the obtained nonlinear global system has a much smaller size, whose Jacobian is symmetric and possibly positive definite. Thus, the residual equation of the approximate Newton approach for the global system can be efficiently solved by standard optimal solvers. As an alternative, we point out that the above two implicit methods combined with operator splittings can also efficiently solve the bidomain equations. Numerical results show that the CBDF2 scheme is an efficient time integration method while achieving high stability and accuracy.