Time Step Increment Research Articles

Particle Virtual Element Method (PVEM): an agglomeration technique for mesh optimization in explicit Lagrangian free-surface fluid modelling

Explicit solvers are commonly used for simulating fast dynamic and highly nonlinear engineering problems. However, these solvers are only conditionally stable, requiring very small time-step increments determined by the characteristic length of the smallest, and often most distorted, element in the mesh. In the Lagrangian description of fluid motion, the computational mesh quickly deteriorates. To circumvent this problem, the Particle Finite Element Method (PFEM) creates a new mesh (e.g., through a Delaunay tessellation, based on node positions) when the current one becomes overly distorted. A fast and efficient remeshing technique is therefore of pivotal importance for an effective PFEM implementation in explicit dynamics. Unfortunately, the 3D Delaunay tessellation does not guarantee well-shaped elements, often generating zero- or near-zero-volume elements (slivers), which drastically reduce the stable time-step size. Available mesh optimization techniques have limited applicability due to their high computational cost when runtime remeshing is required. An innovative possibility to overcome this problem is the use of the Virtual Element Method (VEM), a variant of the finite element method that can make use of polyhedral elements of arbitrary shapes and number of nodes. This paper presents the formulation of a 3D first-order Particle Virtual Element Method (PVEM) for weakly compressible flows. Starting from a tetrahedral mesh, poorly shaped elements, such as slivers, are agglomerated to form polyhedral Virtual Elements (VEs) with a controlled characteristic length. This approach ensures full control over the minimum time-step size in explicit dynamics simulations, maintaining stability throughout the entire analysis.

Physics‐constrained symbolic model discovery for polyconvex incompressible hyperelastic materials

AbstractWe present a machine learning framework capable of consistently inferring mathematical expressions of hyperelastic energy functionals for incompressible materials from sparse experimental data and physical laws. To achieve this goal, we propose a polyconvex neural additive model (PNAM) that enables us to express the hyperelastic model in a learnable feature space while enforcing polyconvexity. An upshot of this feature space obtained via the PNAM is that (1) it is spanned by a set of univariate basis functions that can be re‐parametrized with a more complex mathematical form, and (2) the resultant elasticity model is guaranteed to fulfill the polyconvexity, which ensures that the acoustic tensor remains elliptic for any deformation. To further improve the interpretability, we use genetic programming to convert each univariate basis into a compact mathematical expression. The resultant multi‐variable mathematical models obtained from this proposed framework are not only more interpretable but are also proven to fulfill physical laws. By controlling the compactness of the learned symbolic form, the machine learning‐generated mathematical model also requires fewer arithmetic operations than its deep neural network counterparts during deployment. This latter attribute is crucial for scaling large‐scale simulations where the constitutive responses of every integration point must be updated within each incremental time step. We compare our proposed model discovery framework against other state‐of‐the‐art alternatives to assess the robustness and efficiency of the training algorithms and examine the trade‐off between interpretability, accuracy, and precision of the learned symbolic hyperelastic models obtained from different approaches. Our numerical results suggest that our approach extrapolates well outside the training data regime due to the precise incorporation of physics‐based knowledge.

A multi-resolution DFPM-PD model for efficient solution of FSI problems with structural deformation and failure

This paper presents an accurate and efficient multi-resolution model for reproductions of fluid-structure interaction (FSI) issues involving intricate fluid flow phenomena, as well as structural deformation and failure. The proposed model comprises two modules, i.e., the decoupled finite particle method (DFPM) module for the solution of fluid flow, and the peridynamics (PD) module for simulating motion, deformation, fracture, and contact of structures. By employing distinct particle spacing and time step increments for the two modules, the model achieves spatial-temporal multi-resolution discretization. To address the fluid-structure interfacial region accurately, a disguised ghost particle (DGP) coupling scheme, incorporating a common smoothing length and a tensile instability control (TIC) technique, is developed. The particle inconsistency at the fluid-structure interface is resolved through the use of ghost particles and DFPM approximation, thus enabling more precise computation of interaction forces. Furthermore, a contact model is incorporated into the PD module to reflect contact interactions between solid structures. A set of benchmarks are investigated to validate the accuracy, robustness as well as efficiency of the presented multi-resolution model, and then it is further employed to simulate structural failure due to tsunami waves, which demonstrates its capability of handling engineering FSI problems with structural failure.

A hybrid implicit-explicit discontinuous Galerkin spectral element time domain (DG-SETD) method for computational elastodynamics

SUMMARY A hybrid implicit-explicit (IMEX) discontinuous Galerkin spectral element time domain (DG-SETD) algorithm is proposed to simulate 3D elastic wave propagation in inhomogeneous media. In this method, the original problem can be divided into a number of well designed subdomains, and the mesh generation of different subdomains is completely independent, thus allowing flexible spatial discretization of complex geometry. The neighboring subdomains are connected by a Riemann transmission condition (RTC), and spectral elements with different interpolation orders can be used in different subdomains to maximize the computational efficiency of multiscale problems to facilitate parallel computing for different subdomains. In particular, the explicit or implicit time iteration scheme can be appropriately selected for a subdomain according to the size of its mesh elements to increase the global time step increment, thus giving higher computational efficiency: For subdomains with coarse meshes, the explicit time integration scheme is used and the time step increment is limited by the Courant−Friedrichs−Lewy (CFL) stability condition; for subdomain with fine structures and thus fine meshes, an implicit time integration scheme is used so that a large time step increment can be used without affecting the stability. In addition, a jump condition of displacement and velocity is introduced to accurately simulating fractures and faults, including lossless and viscous fractures with plane, curve or cross structures. This avoids the volume modeling of the extremely thin fracture structures, and effectively reduces the number of degrees of freedoms (DoFs) in the discretized system without the loss of accuracy. The accuracy, robustness and efficiency of the DG-SETD algorithm are demonstrated by quantitative comparisons of the waveforms with the commercial finite element software COMSOL.

Parametric optimization of novel solar chimney power plant using response surface methodology

CFD and experimental approaches are used to carry out the design of a Solar Chimney Power Plant (SCPP) that consists of a Semi Convergent Collector that is equipped with a Divergent Chimney. The present work engages in figuring out the importance of study variables in statistic modelling thereby improving the efficiency. Both the response surface methodology (RSM) and the single objective approach (SOA) are used in the process of determining the amounts of influential parameters and optimizing the model. Experiment test on a small-scaled model is carried out with inclined Collector to a Divergent Chimney. The CFD result indicates that the updraft effect increases because of the convergent collector with mean temperature rise of 18 K. Experiment and simulation results are in good agreement. The numerical model predicted that the increment in outside temperature leads to increment in mass flow rate, velocity, inside temperature, power output and efficiency of the plant. Hence the single input model predicted the values for five outputs. The numerical model seems to be fit with experimental study with an increase in mass flow rate from 0.04 kg/s to 0.06 kg/s with incremental time step. The deviation between the experimental and numerical studies lies only 0.423. Developing a larger size model helps in achieving larger power output. It is recommended that the semi convergent collector with a divergent chimney adopts for construction of large size SCPP for generating greener energy.

An improved method to estimate the rate of change of streamflow recession and basin synthetic recession parameters from hydrographs

Recession analysis of -dQ/dt ∼ Q is popularly used for determining catchment storage-discharge relationship, estimating the aquifer hydraulic properties, and predicting low flow processes. However, there are uncertainties arising from the numerical approximation of -dQ⁄dt by the finite difference (ΔQ/Δt) for individual recession and the fitting cloud-like data of -dQ⁄dt ∼ Q for many recession events in a catchment. In this study, we proposed an improved variable time step increment method (IVTS) to minimize the effects of the noisy observations on estimating -dQ⁄dt, and a fitting method in terms of the binning-average of -dQ⁄dt ∼ Q data points to determine the basin synthetic recession parameters. The proposed methods are validated by using numerical generation of recessions with different noises and recession behaviors and the observed hydrographs from twenty catchments in Huai River Basin in China. The results from our proposed method are compared with those from the other popularly used methods. The IVTS method is proven to be robust for approximation of -dQ⁄dt, which can significantly reduce errors of the estimated -dQ⁄dt and the recession slope (b). Combining with the IVTS method, the fitting method in terms of the binning average by splitting range of -dQ⁄dt, instead of Q, could overcome the shortage of traditional recession analysis methods that underestimate the recession slope. Our study indicates that the more accurate and reliable calculation of -dQ⁄dt ∼ Q data and estimation of basin synthetic recession parameters could increase the objectivity in interpretation of recession behaviors and accuracy in perdition of low flow processes.

Dynamic behaviour of optimal portfolio with stochastic volatility

In the existing literature, little is known about the dynamic behaviour of the optimal portfolio in terms of market inputs and arbitrary stochastic factor dynamics in an incomplete market with a stochastic volatility. In this paper, to study optimal portfolio behaviour, we compute and analyze the mean and the variance of the optimal portfolio and of their adjustment speed in terms of market inputs in an incomplete market. The incompleteness arises from the additional source of uncertainty of the volatility in Heston’s stochastic volatility model. Conducting sensitivity analysis for the mean and the variance of the optimal portfolio process as well as its adjustment speed to the market parameters, we find several interesting behavioural patterns of investors towards asset price and its volatility shocks. Our results are robust and convergent by the agreement from two simulation methods for different time step increments and the number of Monte Carlo simulation paths.

On dissipative gradient effect in higher-order strain gradient plasticity: the modelling of surface passivation

The phenomenological flow theory of higher-order strain gradient plasticity proposed by Fleck and Hutchinson (J. Mech. Phys. Solids, 2001) and then improved by Fleck and Willis (J. Mech. Phys. Solids, 2009) is used to investigate the surface-passivation problem and micro-scale plasticity. An extremum principle is stated for the theory involving one material length scale. To solve the initial boundary value problem, a numerical scheme based on the framework of variational constitutive updates is developed for the strain gradient plasticity theory. The main idea is that, in each incremental time step, the value of the effective plastic strain is obtained through the variation of a functional in regard to effective plastic strain, provided the displacement or deformation gradient. Numerical results for elasto-plastic foils under tension and bending, thin wires under torsion, are given by using the minimum principle and the numerical scheme. Implications for the role of dissipative gradient effect are explored for three non-proportional loading conditions: (1) stretch-passivation problem, (2) bending-passivation problem, and (3) torsion-passivation problem. The results indicate that, within the Fleck–Hutchinson–Willis theory, the dissipative length scale controls the strengthening size effect, i.e. the increase of initial yielding strength, while the surface passivation gives rise to an increase of strain hardening rate.

3-D FDTD Modeling of Electromagnetic Wave Propagation in Magnetized Plasma Requiring Singular Updates to the Current Density Equation

A new finite-difference time-domain (FDTD) algorithm for electromagnetic wave propagation in magnetized plasma is proposed. This algorithm permits the use of two time step increments: one for Maxwell’s equations, ${\Delta t}$ , and the other for the current density equation derived from the Lorentz equation of motion, ${\Delta }{t}_{c}$ . A major advantage of this algorithm over previous approaches is that only a single update iteration is needed for the current density equation even when ${\Delta }{t}_{c} . This provides significant time savings that can make previously infeasibly long simulations now practical. The algorithm’s implementation is also relatively simple and it has relatively low memory requirements. The algorithm is validated against analytical results. A stability analysis is performed.

Variable Damping Profiles Using Modal Analysis for Laser Shock Peening Simulation

The single explicit analysis using time-dependent damping (SEATD) technique for laser shock peening (LSP) simulation employs variable damping to relax the excited model between laser shots, thus distinguishing it from conventional optimum constant damping methods. Dynamic relaxation (DR) is the well-established conventional technique that mathematically identifies the optimum constant damping coefficient and incremental time-step that guarantees stability and convergence while damping all mode shapes uniformly when bringing a model to quasi-static equilibrium. Examined in this research is a new systematic procedure to strive for a more effective, time-dependent variable damping profile for general LSP configurations and boundary conditions, based on excited modal parameters of a given laser-shocked system. The effects of increasing the number of mode shapes and selecting modes by contributed effective masses are studied, and a procedure to identify the most efficient variable damping profile is designed. Two different simulation cases are studied. It is found that the computational time is reduced by up to 25% (62.5 min) for just five laser shots using the presented variable damping method versus conventional optimum constant damping. Since LSP typically involved hundreds of shots, the accumulated savings in computation time during prediction of desired process parameters is significant.

Analyzing Graphene-Based Absorber by Using the WCS-FDTD Method

A weakly conditionally stable finite-difference time-domain (WCS-FDTD) method is used to simulate a graphene-based absorber. By using the auxiliary differential equation and Pade fitting method, both the interband and intraband conductivities of the graphene are incorporated into the WCS-FDTD method. The time step increment in the proposed method is not determined by the fine meshes in the graphene layer, so the computational efficiency of this method is greatly improved from that of the conventional FDTD method. By using the proposed WCS-FDTD method, a graphene-based absorber is simulated and analyzed. The numerical result shows that the graphene can achieve tunable absorption through controlling its chemical potential, and the interband conductivity of the graphene has important effects on the performance of the absorber.

Portfolio Selection and Dynamic Behavior in Heston's Stochastic Volatility Model Using a Contingent Claim

In the existing literature, little is known about the dynamic behavior of the optimal portfolio in terms of market inputs and arbitrary stochastic factor dynamics in an incomplete market with a stochastic volatility. In this paper, to study the optimal portfolio behavior, we compute and analyze the mean and the variance of the optimal portfolio and of its adjustment speed in terms of market inputs in an incomplete market. The incompleteness arises from the additional source of uncertainty of the volatility in Heston’s stochastic volatility model. We discuss a novel way of completing the incomplete market by introducing an artificial European option, which is then made completely uninteresting to the investor. We then apply the martingale method to the completed market to obtain the optimal portfolio in a closed form. Conducting sensitivity analysis for the mean and the variance of the optimal portfolio process as well as its adjustment speed to the market parameters, we find several interesting investor’s behavioral patterns toward asset price and its volatility shocks. Our results are robust and convergent by the agreement from two simulation methods for different time step increments and the numbers of Monte Carlo simulation paths.

Granular contact dynamics with elastic bond model

This paper proposes an elastic bond model in the framework of contact dynamics based on mathematic programming. The bond model developed in this paper can be used to model cemented materials. The formulation can be reduced to model pure static problems without introducing any artificial damping. In addition, omitting the elastic terms in the objective function turns the formulation into rigid bond model, which can be used for the modeling of rigid or stiffly bonded materials. The developed bond model has the advantage over the explicit DEM that large time step or displacement increment can be used. The tensile and shear strength criteria of the bond model are formulated based on the modified Mohr–Coulomb failure criterion. The torque transmission of bonds is introduced based on rolling resistance model. The loss of shear or tensile strength, or torque transmission will lead to the breakage of bonds, and turn the bond into purely frictional contact. Three simple examples are first used to validate the bond model. Numerical examples of uniaxial and biaxial compression tests are used to show its potential in modeling cemented geomaterials. Numerical results show that elastic bonds are indeed necessary for the modeling of cemented granular material under static conditions.

Unified Lagrangian formulation for solid and fluid mechanics and FSI problems

We present a Lagrangian monolithic strategy for solving fluid–structure interaction (FSI) problems. The formulation is called Unified because fluids and solids are solved using the same solution scheme and unknown variables. The method is based on a mixed velocity–pressure formulation. Each time step increment is solved via an iterative partitioned two-step procedure. The Particle Finite Element Method (PFEM) is used for solving the fluid parts of the domain, while for the solid ones the Finite Element Method (FEM) is employed. Both velocity and pressure fields are interpolated using linear shape functions. For quasi-incompressible materials, the solution scheme is stabilized via the Finite Calculus (FIC) method. The stabilized elements for quasi-incompressible hypoelastic solids and Newtonian fluids are called VPS/S-element and VPS/F-element, respectively. Other two non-stabilized elements are derived for hypoelastic solids. One is based on a Velocity formulation (V-element) and the other on a mixed Velocity–Pressure scheme (VP-element). The algorithms for coupling the solid elements with the VPS/F fluid element are explained in detail. The Unified formulation is validated by solving benchmark FSI problems and by comparing the numerical solution to the ones published in the literature.

On a mixed time integration procedure for non-linear structural dynamics

Purpose– The purpose of this paper is to report the implementation of an alternative time integration procedure for the dynamic non-linear analysis of structures.Design/methodology/approach– The time integration algorithm discussed in this work corresponds to a spectral decomposition technique implemented in the time domain. As in the case of the modal decomposition in space, the numerical efficiency of the resulting integration scheme depends on the possibility of uncoupling the equations of motion. This is achieved by solving an eigenvalue problem in the time domain that only depends on the approximation basis being implemented. Complete sets of orthogonal Legendre polynomials are used to define the time approximation basis required by the model.Findings– A classical example with known analytical solution is presented to validate the model, in linear and non-linear analysis. The efficiency of the numerical technique is assessed. Comparisons are made with the classical Newmark method applied to the solution of both linear and non-linear dynamics. The mixed time integration technique presents some interesting features making very attractive its application to the analysis of non-linear dynamic systems. It corresponds in essence to a modal decomposition technique implemented in the time domain. As in the case of the modal decomposition in space, the numerical efficiency of the resulting integration scheme depends on the possibility of uncoupling the equations of motion.Originality/value– One of the main advantages of this technique is the possibility of considering relatively large time step increments which enhances the computational efficiency of the numerical procedure. Due to its characteristics, this method is well suited to parallel processing, one of the features that have to be conveniently explored in the near future.

A 2-D enlarged cell technique (ECT) for elastic wave modelling on a curved free surface

The conventional finite-difference time-domain (FDTD) method for elastic waves suffers from the staircasing error when applied to model a curved free surface because of the structured grid. This is similar to the situation for the FDTD method in electromagnetics when it is applied to model a curved perfect conductor surface, where the conformal FDTD methods have been recently developed to avoid this error. In this work a stable and second-order accurate 2-D FDTD method for elastic wave modelling on a curved free surface is presented based on the finite volume method and enlarged cell technique (ECT). To achieve a sufficiently accurate implementation, a finite volume scheme is applied to the curved free surface to remove the staircasing error; in the meantime, to achieve the same stability as the FDTD method without reducing the time step increment, the ECT is introduced to preserve the solution stability even for small irregular cells. This method is verified by several 2-D numerical examples. Results show that the method is second-order accurate and stable at the Courant stability limit for a regular FDTD grid.

An Efficient 3-D FDTD Model of Electromagnetic Wave Propagation in Magnetized Plasma

Modeling electromagnetic wave propagation in the upper atmosphere is important for space weather effects, satellite communications ionospheric modification experiments, and many other applications. We propose a new methodology for solving and incorporating the current equation into the finite-difference time-domain (FDTD) form of Maxwell's equations for modeling electromagnetic wave propagation in magnetized plasma. This approach employs a version of Boris's algorithm applied to particle-in-cell plasma computational models. There are four primary advantages of this new method over previously developed three-dimensional FDTD models of electromagnetic wave propagation in magnetized plasma. Specifically, it: 1) requires less memory; 2) is more than 50% faster; 3) is easier to implement; and 4) permits the use of two different time step increments when solving the current equation versus Maxwell's equations that is useful for modeling high collisional regimes. The new algorithm is faster because it solves all the equations explicitly and there is no need to solve complicated matrix equations. Modeling of higher altitude ranges and higher frequency electromagnetic waves is much more feasible using this new method. Results of the new FDTD magnetized plasma model are provided and validated.

Numerical Investigations of an Implicit Leapfrog Time-Domain Meshless Method

Numerical solution of partial differential equations governing time domain simulations in computational electromagnetics, is usually based on grid methods in space and on explicit schemes in time. A predefined grid in the problem domain and a stability step size restriction need. Recently, the authors have reformulated the meshless framework based on smoothed particle hydrodynamics, in order to be applied for time domain electromagnetic simulation. Despite the good spatial properties, the numerical explicit time integration introduces, also in a meshless context, a severe constraint. In this paper, at first, the stability condition is addressed in a general way by allowing the time step increment get away from the minimum points spacing. Then, an alternating direction implicit leapfrog scheme for time evolution is proposed. The unconditional stability of the method is analytically provided and numerically validated. The stability of the method has been proved by avoiding the algebra developments related to the usually adopted von Neumann analysis. Three case studies are investigated by achieving a satisfactory agreement by comparing both numerical and analytical results.

Geometrically nonlinear dynamic analysis of thin shells by a four-node quadrilateral element with in-plane rotational degree of freedom

A simple and effective finite element incremental formulation based on the updated lagrangian corotational description for geometrically nonlinear dynamic analysis of shell structure is presented in this work. The flat shell element used is the classical four-node quadrilateral DKQ shell finite element, combined with the improvements of the in-plane behaviour by incorporation of the drilling rotation degree of freedom. The main goal is to have a good flat shell element of quadrilateral geometry, leading to reliable solutions in linear and geometrically nonlinear dynamic analysis. The Newmark’s direct time integration method is adopted to integrate the equations of motion, while the Newton–Raphson method is used for iterating within each time step increment until equilibrium is achieved. The results obtained through a series of selected examples demonstrate the effectiveness of the used shell finite element to predict the nonlinear dynamic response of complex structures, and the robustness of this nonlinea...

Discrete crack analysis of concrete gravity dams based on the known inertia force field of linear response analysis

This paper presents a two-step approach for discrete crack analysis of concrete gravity dams under earthquake force. In this approach, the time-varying inertia forces in a dam are first obtained by linear response analysis. Then, for each time-step increment a discrete crack analysis of the dam is performed under the known force condition. This two-step approach transforms the seismic crack analysis of dams from dynamic analysis to static analysis, based on the intuitive conjecture that the effect of cracks on structural acceleration in gravity dams is small, thus allowing the actual inertia force (the product of mass and acceleration) to be approximately obtained by linear response analysis. This conjecture was proved, and numerical studies showed the strength of the method in tracing discrete cracking behaviours of a dam during large earthquakes. A mathematical generalisation of the solution strategy is also presented to enable the method to be applied to other nonlinear response problems that do not have exact solutions due to various mathematical difficulties in their solution processes.