Time Integration Procedure Research Articles

A lattice modelling framework for fracture-induced acoustic emission wave propagation in concrete

To date, there is no comprehensive approach available that can explicitly model the complete transient waveforms of acoustic emissions (AE) induced by fracture processes in brittle and quasi-brittle materials like concrete. The complexity of AE modelling arises from the intricate coupling between the local discontinuity of material fracturing and the global continuity of elastic wave propagation in solids. Among others, the lattice type models are promising approaches, as they are known to be a matured modelling approach to simulate the fracturing process in concrete-like materials. Nevertheless, like other discrete element methods (DEM), they are currently limited to describing the number and rate of AE events (broken elements) in the fracture process and cannot explicitly model wave generation and propagation. In this study, we propose a lattice modeling framework to simulate the propagation of complete waveforms of fracture-induced AE signals in concrete. A proportional-integral-derivative (PID) control algorithm is incorporated in an explicit time integration procedure to reduce dynamic noise from spurious oscillations. Additionally, a Rayleigh damping-based calculation method and corresponding calibration procedure are proposed to model the attenuation of AE signals due to material damping. Using the developed approach, we systematically investigate the feasibility of lattice models for elastic wave propagation simulation, the dependence of lattice mesh sizes and the choice of numerical damping parameters. These results lead to a systematic framework which can be employed in simulating wave propagation with attenuation using DEM models in general including lattice models.

The α-generalized implicit method associated with Potra–Pták iteration for solving non-linear dynamic problems

Generally, direct time integration procedures are used for solving the equations of motion in transient analysis of structures with large displacements. In this context, we propose an algorithm that combines the α-Generalized implicit integration method with the Potra–Pták two-step iterative scheme. The free Scilab program develops a computer code for the non-linear dynamic analysis of plane frames with large displacements and rotations. The FEM corotational formulation discretizes the structures considering the Euler-Bernoulli beam theory. The null-length connection element described by the axial, translational and rotational stiffnesses simulate the behavior of the beam-column connection. Jacobi’s method and the Scilab’s spec function determine the natural frequencies. The developed program is used for modal and transient dynamic analyses of frame problems available in the literature. The numerical results show that the Potra-Pták scheme obtains approximate solutions with fewer cumulative iterations until convergence and shorter processing time compared to the standard Newton-Raphson scheme. Further-more, the results show that the type of beam-column connection affects the vibratory behavior of the structure as well as the values of its natural frequencies.

Numerical simulation and experimental study on effect of cooling rate on microstructure and strength of nanostructured materials

In this study, by numerical simulation (finite element method, FEM) and experimental, the cooling rate was investigated by changing the product thickness (20, 3, 2, 1, 0.5 and 0.3) mm of Al based two-phase nanostructured materials casted through a copper mold. The effect of cooling rate on the microstructure and strength of the alloy was studied. The Experimental results showed that the precipitated intermetallic phases have a decreasing size corresponding to the increasing cooling rates by simulation from ~102 K/s to ~104 K/s. The results show that an appropriate cooling rate can improve the microstructure and properties of the alloy. The Abaqus/Standard capability for uncoupled heat transfer analysis was intended to model solid body heat conduction with general, temperature-dependent conductivity; internal energy (including latent heat effects); and quite general convection and radiation boundary conditions. This study describes the basic energy balance, constitutive models, boundary conditions, finite element discretization, and time integration procedures used. The time step used an automatic algorithm through the smallest tolerance. The maximum temperature change was allowed over a period and the increment was adjusted for this parameter, as was the rate of convergence in the non-linear cases. First-order heat transfer elements used the rule of numerical integration with integrated stations located at the corners of the element for thermal capacitance terms. (Jacobian terminology). This approach is particularly effective when there is a strong latent thermal effect. Thus, first - order elements were used in the case of latent heat. The HEATCAP element is available for single - point pooled thermal capacitance modeling. Centralized film loading options between the mold and the casting were specified by the user.

A Numerical Time Integration Procedure for Secondary Dendrite Arm Spacing in Hyper-Peritectic Steel Alloys

AbstractA numerical time integration procedure for the calculation of the secondary dendrite arm spacing (SDAS) in FeC hyper-peritectic alloys is presented, a preferred group of low-carbon casting steel. The procedure incorporates a three-stage thin arm dissolution model, which is solved at each time step using Newton–Raphson iteration. This is coupled to a coarsening model based on the integral forms of the dissolution model, which are solved using Gaussian quadrature, as well as a growth model for solid fraction evolution. The procedure can easily be embedded into numerical models for solidification, in which the space–time evolution of the SDAS is required for determining the dynamic permeability in the mushy zone. Higher order approximations for both growth and solute concentration evolution can easily be incorporated. Temperature dependence of thermophysical parameters is taken into account using inter-dendritic solidification empirical models, and an alloy-specific peritectic reaction constant is used to determine the isothermal peritectic holding time. The procedure is validated against experimental data presented in literature. Various cases of SDAS as a function of local solidification time, cooling rate and carbon composition are investigated. The method is compared to experimental results of SDAS obtained from test castings of a hyper-peritectic steel alloy and can be used to iteratively determine the alloy-specific peritectic reaction constant by comparing the solid fraction evolution during the peritectic reaction with that found from the experimental cooling curve.

A material/element-defined time integration procedure for dynamic analysis

A material/element-defined time integration procedure for dynamic analysis

Concurrent Topology Optimization of Multi-Scale Composite Structures Subjected to Dynamic Loads in the Time Domain

This paper presents a concurrent topology optimization of multi-scale composite structures subjected to general time-dependent loads for minimizing dynamic compliance. A three-field density-based method is adopted to implement the concurrent topological design, with macroscopic effective properties of the microstructure evaluated through energy-based homogenization method (EBHM). Transient response is obtained from the two-scale finite element analysis with the HHT-α approach as an implicit time integration procedure. Design sensitivities are formulated employing the adjoint variable method (AVM) based on two main philosophies: “discretize-then-differentiate” and “differentiate-then-discretize” approaches, respectively. The method of moving asymptotes is adopted to update the design variables at two scales. Several benchmark examples are presented to demonstrate that the “discretize-then-differentiate” AVM attains consistent sensitivities in an inherent manner such that the resulting optimal topology is more efficient when compared with the “differentiate-then-discretize” AVM. Moreover, the potential of the proposed method for concurrent dynamic topology optimization problems under general time-dependent loads is also highlighted.

Formulation and Analysis of a Cubic B-Spline-Based Time Integration Procedure for Structural Seismic Response Analysis

In this study, an explicit time integration scheme based on cubic B-spline interpolation strategy and momentum corrector is extended for structural seismic response analysis. The acquisition strategy for the required seismic load is provided. The cubic B-spline-based scheme with moment corrector is generalized for the nonlinear hysteresis system. The analysis shows that the explicit scheme has more desirable stability and accuracy properties than other competitive schemes. The effectiveness of the scheme is demonstrated by the linear and nonlinear numerical examples.

Higher-order aerodynamic numerical simulations in compressible RANS framework with inverse-ω scale variable

The use of higher-order methods to robustly compute turbulent flows governed by the multi-equation turbulence model in Reynolds-averaged Navier-Stokes (RANS) framework is a challenging topic in the computational fluid dynamics (CFD) community. In this paper, we propose two aspects of measures involving the physical model and the numerical scheme. Firstly, a λ-scale (=1/ωn) is derived instead of the well-known ω in SSG/LRR Reynolds-stress model (RSM) and SST eddy viscosity model (EVM) due to its natural boundary conditions for viscous walls and help to the positivity preservation of dissipation rate ε in turbulence equations when n=1/(2m),m=1,2,3.... Secondly, a geometric-conservation-preservation negativity-correction (GCPNC) method is proposed on the implicit time integration procedure with right-hand-side (RHS) term discretized by high-order weighted compact nonlinear schemes (WCNSs). It can guarantee the positivity of normal Reynolds stresses or turbulent kinetic energy. Numerical tests with real-life compressible flows are reported to demonstrate the robustness and accuracy of the present methods. Relevant cases include the supersonic square duct flow, the flows over delta wing, wing-body configuration and high-lift configuration.

Virtual modelling technique for geometric-material nonlinear dynamics of structures

This paper presents a virtual modelling technique for dynamic safety assessment of practical structures undergoing geometric and material blended nonlinearities. The variational inputs of systematic properties are treated within the 3D dynamic geometric-elastoplastic analyses. To circumvent numerical challenges in solving the coupled nonlinear problems, a freshly developed dynamic virtual modelling (DVM) technique is employed to determine the inherent relationship between the variational input data and the nonlinear structural response by using a new clustering based extended support vector regression (C-XSVR) algorithm with a novel T-spline polynomial kernel function. The virtual modelling models can be constructed at each time step within the Newmark’s time integration procedure, which then can be used to predict deflection, force, and stress of the concerned structure at different periods. The DVM is capable of visibly forecasting potential large deformation nonlinear behaviours in an efficient manner, based on the explicit relationship functions. To demonstrate the accuracy and effectiveness of the proposed framework, nonlinear behaviours of two practical applications under future forecasted working conditions are predicted and validated in the numerical investigations.

Simulation of non-linear structural elastodynamic and impact problems using minimum energy high-order bases

We present the application of simultaneous diagonalization and minimum energy (SDME) high-order finite element modal bases for simulation of transient non-linear elastodynamic problem, including impact cases with Hookean and neo-Hookean hyperelastic materials. The bases are constructed using procedures for simultaneous diagonalization of the internal modes and Schur complement of the boundary modes from the standard nodal and modal bases, constructed using Lagrange and Jacobi polynomials, respectively. The implementation of these bases in a high-order finite element code is straightforward, since the procedure is applied only to the one-dimensional expansion bases. Non-linear transient structural problems with large deformation , hyperelastic materials and impact are solved using the obtained bases with explicit and implicit time integration procedures. Iterative solutions based on preconditioned conjugate gradient methods are considered. The performance of the proposed bases in terms of the number of iterations of pre-conditioned conjugate gradient methods and computational time are compared with the standard nodal and modal bases. The SDME bases are accurate and provide better conditioned equations for iterative solutions in general, leading to substantial reduction in processing time over an order of magnitude compared to other modal bases. Our numerical tests obtained speedups up to 41 using the considered bases when compared to the standard modal basis. Results for a two-dimensional impact problem approximated with the SDME bases showed that the implicit Newmark method performed much better in terms of processing time and contact stress error when compared to the standard explicit central difference method with lumped mass matrix . • Application of simultaneous diagonalization procedure and minimum energy highorder bases to non-linear structural and impact transient problems. • Simple modifications of high-order bases improve conditioning of system of equations obtained from the high-order approximation of the considered problems, with remarkable gains in performance of iterative solvers. • Light variation on the number of pre-conditioned gradient iterations in terms of the polynomial order. • Speed-ups over 40 when compared to the standard modal Jacobi high-order bases used in high-order interpolation.

An improved adaptive formulation for explicit analyses of wave propagation models considering locally-defined self-adjustable time-integration parameters

In this paper, a novel explicit time-marching procedure is proposed for wave propagation analyses, considering an improved adaptive formulation. In this adaptive approach, different time-integration parameters are defined for each element of the adopted spatial discretization, allowing different stability limits (as well as other properties) to be locally applied. Thus, self-adjustable dissipative and non-dissipative parameters are automatically distributed along the discretized domain of the model, following the properties of its elements, and their use is triggered regarding the behaviour of the computed responses. In this context, the errors of the adopted spatial discretization method may be properly countervailed by the time-integration procedure, allowing substantially better accuracy to be provided. In addition, as is discussed along the manuscript, the novel time-marching formulation is very simple to implement and to apply, and it provides more efficient analyses than standard time integration techniques, enabling larger time-step values to be considered as well as reduced computational efforts to be attained per time step. At the end of the paper, numerical results are presented, illustrating the effectiveness and excellent performance of the proposed technique.

Performance analysis of relaxation Runge–Kutta methods

Recently, global and local relaxation Runge–Kutta methods have been developed for guaranteeing the conservation, dissipation, or other solution properties for general convex functionals whose dynamics are crucial for an ordinary differential equation solution. These novel time integration procedures have an application in a wide range of problems that require dynamics-consistent and stable numerical methods. The application of a relaxation scheme involves solving scalar nonlinear algebraic equations to find the relaxation parameter. Even though root-finding may seem to be a problem technically straightforward and computationally insignificant, we address the problem at scale as we solve full-scale industrial problems on a CPU-powered supercomputer and show its cost to be considerable. In particular, we apply the relaxation schemes in the context of the compressible Navier–Stokes equations and use them to enforce the correct entropy evolution. We use seven different algorithms to solve for the global and local relaxation parameters and analyze their strong scalability. As a result of this analysis, within the global relaxation scheme, we recommend using Brent’s method for problems with a low polynomial degree and of small sizes for the global relaxation scheme, while secant proves to be the best choice for higher polynomial degree solutions and large problem sizes. For the local relaxation scheme, we recommend secant. Further, we compare the schemes’ performance using their most efficient implementations, where we look at their effect on the timestep size, overhead, and weak scalability. We show the global relaxation scheme to be always more expensive than the local approach—typically 1.1–1.5 times the cost. At the same time, we highlight scenarios where the global relaxation scheme might underperform due to its increased communication requirements. Finally, we present an analysis that sets expectations on the computational overhead anticipated based on the system properties.

RETRACTED ARTICLE: An Implicit Unconditionally Stable Integration Method for Nonlinear Structural Dynamics and Earthquake Engineering Problems

A novel time integration procedure is designed in order to solve the differential equation of motion of dynamics and earthquake engineering problems. The procedure is constituted on the principle of impulse-momentum, leading to a lesser number of assumed fields. The algorithmic properties of the procedure are determined by stability and accuracy analyses. Overshooting tendency, which is not related to stability and accuracy characteristic of a method, and order of accuracy are also examined. It is displayed that the new method is unconditionally stable and non-dissipative. Also, the numerical dispersion of the proposed algorithm appears to be much less than the commonly used integration methods. The method has no tendency to overshoot both the displacement and velocity response solutions. Its order of accuracy is around four as compared to two of the other methods considered in the study. A few numerical examples consisting of both single and multi-degree of freedom systems with linear and nonlinear characteristics are performed to see the overall behavior of the method in various practical problems. The numerical results of the proposed method obtained from these examples coincide well with the simulated exact results.

Mode-displacement method for structural dynamic analysis of bio-inspired structures: A palm-tree stem subject to wind effects

ABSTRACT Biological materials (orthotropic materials), like wood, can offer good mechanical properties with a minimum amount of material, making their internal structure the suitable one to be applied on bio-inspired structures. The knowledge of the exceptional structural performance of palm trees, and specially its response to different loading conditions, provides useful information when lightweight structures with high slenderness ratio are desired. Recent researches focused on the analysis of palm trees subject to static loading conditions, ignoring the fluctuating nature of the wind speed. The purpose of this study is to simulate in a computational efficient way the effect of dynamic loading conditions applied on palm trees. Using the mode displacement method, the number of degrees of freedom of a dynamic finite element analysis can be drastically reduced with a minimal loss of accuracy. It was applied to simulate the behavior of structures comprised of an orthotropic material subject to a stochastic dynamic load. The influence of the number of selected degrees of freedom has also been studied. In addition, an exponential integration method is proposed to perform the time integration procedure. The results obtained show that a properly reduced model suitably represents the full finite element model without any appreciable loss of accuracy; it is also shown that computational cost can be drastically reduced. This method could give an appropriate computational representation of the behavior of orthotropic structures, and it could be used for studying more complex bio-inspired structures.

Localized MQ-RBF meshless techniques for modeling unsaturated flow

In this study, we focus on space-time mesh-free numerical techniques for efficiently solving the Richards equation which is often used to model unsaturated flow through porous media. We propose an efficient approach which combines the use of local multiquadric (MQ) radial basis function (RBF) methods and space-time techniques. The localized MQ-RBFs meshless methods allow to avoid mesh generation and ill-conditioning problem where a sparse matrix is obtained for the global system which has the advantage of using reduced memory and computational time. To further reduce the computational cost, we use the space-time techniques having the advantages of solving the resulting algebraic system only once and removing the time-integration procedure. The proposed method has the benefit of considering collocation points on the boundaries of computational domains which makes it more flexible in dealing with complex geometries. We implement the proposed numerical model of infiltration and we perform a series of numerical tests, encompassing various nontrivial solutions, to confirm the performance of the proposed techniques. The numerical simulations show the accuracy, efficiency in terms of computational cost, and capability of the proposed numerical techniques in solving the Richards equation in two-, three- and four-dimensional space-time domains with complex boundaries.

Non-linear dynamic analysis of reinforced concrete structures with hybrid mixed stress finite elements

This paper presents and discusses a hybrid-mixed stress (HMS) finite element model for the static and dynamic analysis of 2D reinforced concrete frame structures. It is assumed a physically non-linear behaviour for both concrete and steel. In this model, both the stress and the displacement fields are approximated in the domain of each element. Complete sets of orthonormal Legendre polynomials are used as approximation functions. The use of these functions makes possible the definition of analytical closed form solutions for the computation of all structural operators when a physically linear behaviour is assumed and leads to the development of very effective p- refinement procedures. Due to the nature of the HMS formulation, it is possible to consider the use of both inverse and direct integration schemes at the cross-section level. The time integration procedure uses classical step-by-step techniques, such as Newmark and α-HHT method. The latter is used in order to supress some numerical noise that may occur in the iterative procedure involved in the solution of dynamic non-linear problems. Three different types of damage models, with and without permanent strains, are used to model concrete behaviour. To validate the model, to illustrate its potential and to assess its accuracy and efficiency, several numerical examples are discussed, and comparisons are made with solutions provided by other numerical techniques.

Performance of a TMD to Mitigate Wind-Induced Interference Effects between Two Industrial Chimneys

The present paper studies the performance of a tuned mass damper (TMD) installed in a 183 m tall chimney located at the edge of the wake shed by another chimney. Numerical and experimental results are available. For the simulations, wind action is considered by solving several 2D flow problems on a selected number of horizontal planes, in the transverse direction to the stacks. On such planes, Navier-Stokes equations are solved to estimate the fluid action at different positions of the chimneys and standard interpolation techniques are applied in the vertical direction. An Arbitrary Lagrangian-Eulerian (ALE) approach is used to consider the moving domain, and a fractional-step scheme is used to solve the fluid field. For the structural modelling, chimneys are meshed using 3D beam finite elements. The time integration procedure used for the structural dynamics is based on the standard second order Bossak method. For each period of time, the fluid problem is solved, the aeroelastic analysis is carried out and the geometry of the fluid mesh of each plane is updated according to the structural movements. With this procedure and model updating techniques, the response of the leeward chimney is evaluated for different scenarios, revealing an interesting dependence of the TMD performance on the wind speed and direction.

Mathematical analysis of the spatial coupling of an explicit temporal adaptive integration scheme with an implicit time integration scheme

AbstractThe Reynolds‐averaged Navier–Stokes equations and the large eddy simulation equations can be coupled using a transition function to switch from a set of equations applied in some areas of a domain to the other set in the other part of the domain. Following this idea, different time integration schemes can be coupled. In this context, we developed a hybrid time integration scheme that spatially couples the explicit scheme of Heun and Crank–Nicolson's implicit scheme using a dedicated transition function. This scheme is linearly stable and second‐order accurate. In this article, an extension of this hybrid scheme is introduced to deal with a temporal adaptive procedure. The idea is to treat the time integration procedure with unstructured grids as it is performed with Cartesian grids and local mesh refinement. Depending on its characteristic size, each mesh cell is assigned to a rank. And for two cells from two consecutive ranks, the ratio of the associated time steps for time marching the solutions is 2. As a consequence, the cells with the lowest rank iterate more than the other ones to reach the same physical time. In a finite‐volume context, a key ingredient is to keep the conservation property for the interfaces that separate two cells of different ranks. After introducing the different schemes, the article recalls briefly the coupling procedure, and details the extension to the temporal adaptive procedure. The new time integrator is validated with the propagation of 1D wave packet, the Sod's tube, and the advection of 2D vortex.

An adaptive sparse kernel technique in greedy algorithm framework to simulate an anomalous solute transport model

In the current work, an efficient and powerful computational technique is implemented to simulate an anomalous mobile-immobile solute transport process. The process is mathematically modelled as a time-fractional mobile-immobile diffusion equation in the sense of Riemann-Liouville derivative. Firstly, an implicit time integration procedure is used to semi-discretize the model in the time direction. The unconditional stability of the proposed time discretization scheme has been proven. Then an adaptive sparse meshless method has been formulated and implemented to fully discretize the model. In this approach, a kernel-based collocation method is equipped with a greedy sparse approximation procedure to discretize the governing problem on a convenient neighborhood of each data point with acceptable accuracy. Therefore, it leads to a sparse and well-conditioned algebraic system. Some test problems on regular and irregular computational domains are presented to verify the validity, efficiency, and accuracy of the method.

Geometrically exact planar Euler-Bernoulli beam and time integration procedure for multibody dynamics

A new formulation of geometrically exact planar Euler-Bernoulli beam in multi-body dynamics is proposed. For many applications, the use of the Euler-Bernoulli model is sufficient and has the advantage of being a nodal displacement-only formulation avoiding the integration of rotational degrees of freedom. In this paper, an energy momentum method is proposed for the nonlinear in-plane dynamics of flexible multi-body systems, including the effects of revolute joints with or without torsional springs. Large rotational angles of the joints are accurately calculated. Several numerical examples demonstrate the accuracy and the capabilities of the new formulation.