Implicit Time Discretization Scheme Research Articles

On hyperexponential stabilization of double integrator in continuous and discrete time

A time-varying state feedback is presented that guarantees stability with hyperexponential rate of convergence of all trajectories to the origin for a double integrator system. It is shown that this convergence property is uniform with respect to matched bounded external disturbances. An implicit time-discretization scheme of the closed-loop system is given, which preserves all main properties of the continuous-time counterpart, and in addition has bounded errors with respect to the measurement noises. Based on this discretization, for sampled-and-hold implementation, a modified linear time-varying state feedback is proposed providing an accelerated rate of convergence to the continuous-time plant. The efficiency of the suggested control is illustrated through numeric experiments.

Multirate Partially Explicit Scheme for Multiscale Flow Problems

For time-dependent problems with high-contrast multiscale coefficients, the time step size for explicit methods is affected by the magnitude of the coefficient parameter. With a suitable construction of multiscale space, one can achieve a stable temporal splitting scheme where the time step size is independent of the contrast. Consider the parabolic equation with heterogeneous diffusion parameter, the flow rates vary significantly in different regions due to the high-contrast features of the diffusivity. In this work, we aim to introduce a multirate partially explicit splitting scheme to achieve efficient simulation with the desired accuracy. We first design multiscale subspaces to handle flow with different speed. For the fast flow, we obtain a low-dimensional subspace with respect to the high-diffusive component and adopt an implicit time discretization scheme. The other multiscale subspace will take care of the slow flow, and the corresponding degrees of freedom are treated explicitly. Then a multirate time stepping is introduced for the two parts. The stability of the multirate methods is analyzed for the partially explicit scheme. Moreover, we derive local error estimators corresponding to the two components of the solutions and provide an upper bound of the errors. An adaptive local temporal refinement framework is then proposed to achieve higher computational efficiency. Several numerical tests are presented to demonstrate the performance of the proposed method.

Modified Gradient Flow Method for Solving One-Dimensional Optimal Control Problem Governed by Linear Equality Constraint

This study presents a computational technique developed for solving linearly constraint optimal control problems using the Gradient Flow Method. This proposed method, called the Modified Gradient Flow Method (MGFM), is based on the continuous gradient flow reformulation of constrained optimization problem with three-level implicit time discretization scheme. The three-level splitting parameters for the discretization of the gradient flow equations are such that the sum of the parameters equal to one (\theta1 + \theta2 +\theta3=1). The Linear and quadratic convergence of the scheme were analyzed and were shown to have first order scheme when each parameter exist in the domain [0, 1] and second order when the third parameter equal to one. Numerical experiments were carried out and the results showed that the approach is very effective for handling this class of constrained optimal control problems. It also compared favorably with the analytical solutions and performed better than the existing schemes in terms of convergence and accuracy

Comparison of implicit time-discretization schemes for hybridized discontinuous Galerkin methods

The present study is focused on the application of two families of implicit time-integration schemes for general time-dependent balance laws of convection-diffusion-reaction type discretized by a hybridized discontinuous Galerkin method in space, namely backward differentiation formulas (BDF) and diagonally implicit Runge-Kutta (DIRK) methods. Special attention is devoted to embedded DIRK methods, which allow the incorporation of time step size adaptation algorithms in order to keep the computational effort as low as possible. The properties of the numerical solution, such as its order of convergence, are investigated by means of suitably chosen test cases for a linear convection-diffusion-reaction equation and the nonlinear system of Navier-Stokes equations. For problems considered in this work, the DIRK methods prove to be superior to high-order BDF methods in terms of both stability and accuracy.

Hybrid Discontinuous Galerkin/Finite Volume Method with Subcell Resolution for Shocked Flows

A hybrid discontinuous Galerkin (DG)/finite volume (FV) method under the multistage boundary variation diminishing (BVD) reconstruction paradigm is developed for shocked inviscid and viscous flows. The methodology uses high-order unlimited polynomials and the THINC (which stands for tangent of hyperbola for interface capturing) function as the candidate reconstruction functions to perform FV computation on embedded subcells. The reconstruction scheme on the FV subcells can be designed in a way to keep the resulting approach as compact as the underlying DG method. The reconstruction information based on the modified BVD selection algorithm has been applied as an indicating strategy so that no additional discontinuity indicator is required for the hybrid DG/FV computation. The designed approach has been investigated through the coupling with the explicit and implicit time discretization scheme for both inviscid and viscous flows. Numerical results demonstrate the high-fidelity characteristics of the method by using a much coarser grid to capture both the discontinuity and the smooth flow structure with high resolution.

Finite element analysis of coupled phase-field and thermoelasticity equations at large strains for martensitic phase transformations based on implicit and explicit time discretization schemes

In this paper, a staggered, nonlinear finite element procedure is developed to solve the large-strain based coupled system of time dependent Ginzburg-Landau (GL) and thermoelasticity equations for phase transformations at the nanoscale. Geometrical nonlinearities are included based on the total Lagrangian description where the total deformation gradient is defined as the multiplicative decomposition of elastic, transformational and thermal deformation gradient tensors. The principle of virtual work is utilized to obtain the integral form of the Lagrangian equation of motion whose discretized form is solved using the iterative methods which gives the thermal and transformational deformation gradient tensors. Next, the elastic deformation gradient tensor and the first Piola-Kirchhoff stress are calculated and substituted in the GL equation which gives the phase order parameter. The weighted residual method is used to derive the corresponding finite element form which allows for the phase-dependent surface energy BCs. Explicit and different implicit methods belonging to the generalized trapezoidal family are used for time discretization of the GL equation. Various examples of phase transformations are simulated and discussed. The effect of different time integration schemes are also investigated. The current study allows for a proper modeling of various phase field problems at large and small strains.

A New Numerical Procedure for Vibration Analysis of Beam under Impulse and Multiharmonics Piezoelectric Actuators

The dynamic behavior of structures with piezoelectric patches is governed by partial differential equations with strong singularities. To directly deal with these equations, well adapted numerical procedures are required. In this work, the differential quadrature method (DQM) combined with a regularization procedure for space and implicit scheme for time discretization is used. The DQM is a simple method that can be implemented with few grid points and can give results with a good accuracy. However, the DQM presents some difficulties when applied to partial differential equations involving strong singularities. This is due to the fact that the subsidiaries of the singular functions cannot be straightforwardly discretized by the DQM. A methodological approach based on the regularization procedure is used here to overcome this difficulty and the derivatives of the Dirac-delta function are replaced by regularized smooth functions. Thanks to this regularization, the resulting differential equations can be directly discretized using the DQM. The efficiency and applicability of the proposed approach are demonstrated in the computation of the dynamic behavior of beams for various boundary conditions and excited by impulse and Multiharmonics piezoelectric actuators. The obtained numerical results are well compared to the developed analytical solution.

Implicit time discretization schemes for mixed least-squares finite element formulations

This work is an extension of the ideas in Averweg et al. (2019) with the focus on a detailed investigation of implicit time discretization schemes to model instationary fluid flow, based on the incompressible Navier–Stokes equations, and linear elastodynamic structural behavior. The variational approaches for fluid and solid mechanics are based on a mixed least-squares finite element method. The L2-norm minimization of the residuals of the constructed first-order systems of the governing differential equations is based on two-field stress–velocity (SV) functionals. For the time discretization of the SV-fluid formulation, four different types of implicit integration schemes are investigated, namely the Houbolt method, the Crank–Nicolson method and two explicit, singly diagonally implicit Runge–Kutta methods (ESDIRK). The SV-formulation for the solid is discretized applying the Houbolt method. The presented time integration schemes are validated investigating an unsteady fluid flow and an elastodynamic structural benchmark. Since both (fluid and solid) SV formulations are discretized using conforming finite element spaces in H(div) and H1, respectively, the inherent fulfillment of coupling conditions, when modeling fluid–structure interaction problems, is given a priori. Therefore, the applicability is also examined by two simplified FSI problems for small deformations, in order to represent the main characteristics of the presented approach.

Existence and uniqueness of the entropy solution of a stochastic conservation law with a Q‐Brownian motion

In this paper, we prove the existence and uniqueness of the entropy solution for a first‐order stochastic conservation law with a multiplicative source term involving a ‐Brownian motion. After having defined a measure‐valued weak entropy solution of the stochastic conservation law, we present the Kato inequality, and as a corollary, we deduce the uniqueness of the measure‐valued weak entropy solution, which coincides with the unique weak entropy solution of the problem. The Kato inequality is proved by a doubling of variables method; to that purpose, we prove the existence and the uniqueness of the strong solution of an associated stochastic nonlinear parabolic problem by means of an implicit time discretization scheme; we also prove its convergence to a measure‐valued entropy solution of the stochastic conservation law, which proves the existence of the measure‐valued entropy solution.

Quantum Dynamics with the Parallel Transport Gauge

The dynamics of a closed quantum system is often studied with the direct evolution of the Schrödinger equation. In this paper, we propose that the gauge choice (i.e., degrees of freedom irrelevant to physical observables) of the Schrödinger equation can be generally nonoptimal for numerical simulation. This can limit, and in some cases severely limit, the time step size. We find that the optimal gauge choice is given by a parallel transport formulation. This parallel transport dynamics can be simply interpreted as the dynamics driven by the residual vectors, analogous to those defined in eigenvalue problems in the time-independent setup. The parallel transport dynamics can be derived from a Hamiltonian structure and is thus suitable to be solved using a symplectic and implicit time discretization scheme, such as the implicit midpoint rule, which allows the usage of a large time step and ensures the long time numerical stability. We analyze the parallel transport dynamics in the context of the singularly perturbed linear Schrödinger equation and demonstrate its superior performance in the near adiabatic regime. We demonstrate the effectiveness of our method using numerical results for linear and nonlinear Schrödinger equations, as well as the time-dependent density functional theory (TDDFT) calculations for electrons in a benzene molecule driven by an ultrashort laser pulse.

Crouzeix–Raviart Finite Element Approximation for the Parabolic Obstacle Problem

Abstract We introduce and study a fully discrete nonconforming finite element approximation for a parabolic variational inequality associated with a general obstacle problem. The method comprises of the Crouzeix–Raviart finite element method for space discretization and implicit backward Euler scheme for time discretization. We derive an error estimate of optimal order 𝒪 ⁢ ( h + Δ ⁢ t ) {\mathcal{O}(h+\Delta t)} in a certain energy norm defined precisely in the article. We only assume the realistic regularity u t ∈ L 2 ⁢ ( 0 , T ; L 2 ⁢ ( Ω ) ) {u_{t}\in L^{2}(0,T;L^{2}(\Omega))} and moreover the analysis is performed without any assumptions on the speed of propagation of the free boundary. We present a numerical experiment to illustrate the theoretical order of convergence derived in the article.

Modeling the martensite reorientation and resulting zero/negative thermal expansion of shape memory alloys

Recent experimental investigation shows that macroscopic overall zero/negative thermal expansion (TE) can be obtained through the martensite reorientation of NiTiPd shape memory alloy (SMA). In this work, a microstructure-based theoretical model is established to quantify such phenomenon. Based on the crystallographic symmetry, the microstructures and the coefficients of thermal expansion of the austenite and martensite lattices of NiTiPd SMA are analyzed. A crystal plasticity-type constitutive model at the individual grain level is constructed to describe the martensite transformation (MT) and martensite reorientation (MR) processes, and the evolution of thermal expansion coefficient. The nonlinear constitutive model is linearized by adopting the implicit backward Euler time discretization scheme and an incremental affine stress–strain–temperature relationship is obtained. An incremental self-consistent homogenization scheme is proposed to describe the interaction among the grains and calculate the elastic, inelastic (MT and MR) and thermal deformations of the polycrystalline aggregates. The established model is used to simulate the MR and the resulting zero/negative TE of polycrystalline NiTiPd SMA. It is shown that the observed quantitative features in experiments can be captured reasonably by the model.

Multibody model based estimation of multiple loads and strain field on a vehicle suspension system

Abstract This work proposes an augmented extended Kalman filter based state-input estimator for mechanical systems defined by implicit equations of motion which is then applied to estimate the six wheel center loads and the strain field on a vehicle suspension test rig. Implicit equations of motion typically arise in the definition of flexible multibody models and also in their time resolution, because implicit time-discretization schemes are normally employed to obtain a stable solution. The presented methodology can be applied to such case and analytical expressions are derived for the necessary linearizations, providing the means for a computationally efficient estimation procedure. The six wheel center loads and the strain field on a vehicle suspension system are valuable quantities during the vehicle design phase (e.g. for durability analysis), hence they are often directly measured during elaborate full vehicle testing campaigns. This work demonstrates that a flexible multibody model representation allows to accurately reconstruct the time domain signals of the six loads and of the full strain field, starting from a minimal set of six measured strains, hence providing an appealing alternative to direct measurement methods. The experimental validation on the suspension test rig shows that all estimated quantities can be accurately reconstructed, given that the system simulation model incorporates an adequate level of accuracy.

Implicit time discretization schemes for least‐squares finite element formulations to model incompressible flows

AbstractIn the present contribution different implicit time integration schemes to model incompressible flow problems are investigated. Amongst the tested methods are the fully implicit Newmark, Houbolt and Crank‐Nicolson schemes.The first‐order system for the proposed formulation is given in terms of stresses and velocities as introduced in [1]. Raviart‐Thomas and standard Lagrange interpolation functions are used to approximate the stresses and velocities, respectively.For the comparison of all time discretization schemes, the unsteady two‐dimensional laminar flow around a cylinder at Reynolds number Re = 100 is chosen. For a rating on the accuracy, the obtained results for the drag coefficient are benchmarked against reference solutions by [2].

Efficient methods with higher order interpolation and MOOD strategy for compressible turbulence simulations

Efficient higher order interpolation schemes based on a multi-dimensional optimal order detection (MOOD) paradigm are developed coupled with the implicit time discretization scheme and further investigated for implicit large eddy simulation of compressible turbulence. The developed methodology utilizes higher-order either the upwind or the central interpolation to minimize numerical dissipation and meantime hybridizes shock capturing scheme that is also provided with higher-order interpolation to stabilize the solution. Simple and effective technique is proposed to apply the current method to an unsteady dual-time stepping scheme, which is to the best of the authors' knowledge the first time to develop the strategy for the MOOD application within an unsteady implicit time discretization framework. The resulting schemes are implemented in the simplified finite volume method that is constructed on non-uniform, curvilinear, multiblock structured grids. Numerical results for a comprehensive suite of both benchmark and practical problems demonstrate that the designed schemes simultaneously obtain the well-resolved broadband turbulence and the sharp shock profiles with considerable reduction in the computation cost.

ADI schemes for valuing European options under the Bates model

This paper is concerned with the adaptation of alternating direction implicit (ADI) time discretization schemes for the numerical solution of partial integro-differential equations (PIDEs) with application to the Bates model in finance. Three different adaptations are formulated and their (von Neumann) stability is analyzed. Ample numerical experiments are provided for the Bates PIDE, illustrating the actual stability and convergence behaviour of the three adaptations.

Verification and validation of a coupled elasto-acoustic-damage system

A novel time-domain method for simulating dynamic damage evolution in a coupled structural-acoustic system is presented. The system is derived via the theory of continuum damage mechanics, and incorporates standard damage evolution models, but is readily extendible to more complex formulations. The overall solution method is staggered, first solving for the dynamic damage evolution with an explicit step, and then using the new values of damage in the coupled computation of the structural-acoustic system. The spatial domain is discretized using a mixed finite element method, and the temporal domain is discretized with a higher-order implicit time discretization scheme. Code and solution verification of the fully coupled solution algorithm are presented, as are validation studies for cases with and without evolving damage. Applications with evolving damage are presented, and present a novel first principles study of changes in the structural acoustic response to dynamically evolving damage in the structure....

Energy Stability Analysis of Some Fully Discrete Numerical Schemes for Incompressible Navier–Stokes Equations on Staggered Grids

In this paper we consider the energy stability estimates for some fully discrete schemes which both consider time and spatial discretizations for the incompressible Navier–Stokes equations. We focus on three kinds of fully discrete schemes, i.e., the linear implicit scheme for time discretization with the finite difference method (FDM) on staggered grids for spatial discretization, pressure-correction schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations, and pressure-stabilization schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations. The energy stability estimates are obtained for the above each fully discrete scheme. The upwind scheme is used in the discretization of the convection term which plays an important role in the design of unconditionally stable discrete schemes. Numerical results are given to verify the theoretical analysis.

A 2-D transient numerical heat transfer model of the solar absorber plate to improve PV/T solar collector systems

This paper presents a genuine 2-D transient numerical model to evaluate the temperature field and the dependent electrical performances of different solar absorber plates. It can readily be adapted for solar thermal absorber plates, standard PV plates or thermally enhanced PV plates that can be used in PV/T solar collectors. The model can also be combined with the appropriate heat exchanger equations to simulate a complete solar collector. A simple optical model is combined with the well-known five parameters PV module electrical model and the 2-D heat conduction equation to compute the temperature field of the PV plate and the electrical power generated for every time step of the simulation. The resulting differential equation is solved through finite volume discretization using an implicit time discretization scheme. The model relies on widely available weather parameters such as total horizontal solar radiation, ambient temperature, wind speed, etc. Validation of the electrical model is first carried out in real outdoor weather conditions comparing simulated and measured electrical current. During this process, measured temperature of the plate and incident solar radiation are used as inputs. Then, the global multi-physic model is validated using a temperature measurement at a selected point and the measured electrical performances. In the multi-physic model, measurements of the incident solar radiation, air temperature, wind speed and relative humidity are used as inputs. The validation results for the temperature showed a maximum discrepancy of the numerical predictions with respect to the measurements of less than 2 [°C] and a discrepancy below 7 [%] for the electric power production both over a complete day of simulation.

Simulation of coupled structural-acoustic response with dynamic damage evolution

A novel time-domain method for simulating dynamic damage evolution in a coupled structural-acoustic system is presented. The system is derived via the theory of continuum damage mechanics, and incorporates standard damage evolution models, but is readily extendible to more exotic formulations. The overall solution method is staggered, solving for the dynamic damage evolution first with an explicit step, and then using the new values in the coupled computation of the structural-acoustic system. The spatial domain is discretized using a mixed finite element method, and the temporal domain is discretized with a higher-order implicit time discretization scheme. Efforts toward fully coupled verification of the solution algorithm are presented, as are validation studies for cases without evolving damage. Applications with evolving damage are presented, and present a novel first principles study of changes in the structural acoustic response to dynamically evolving damage in the structure. Special attention is given to brittle fracture. Examples of downstream usage of the evolving structural response are discussed in the concluding remarks.