Pseudo-transient Continuation Research Articles

Design and optimization of reaction-separation-recycle systems using a pseudo-transient continuation model

The reaction-separation-recycle (RSR) systems are an important part of chemical processes. Due to the interaction between reaction and separation sections, the behavior of RSR processes becomes highly complex and non-linear, posing different challenges for their design and optimization. The purpose of this study is to design and optimize RSR processes for irreversible liquid phase reaction systems using the pseudo-transient continuation (PTC) approach within an equation-oriented programing environment. This approach was applied to investigate one binary system and four ternary systems. The differential-algebraic models of the proposed process flowsheets were solved until the steady-state conditions were reached. The tray bypass efficiency method was incorporated into the PTC to circumvent the need for discrete optimization. The results demonstrated that in those cases where the product was heavier than the reactants, employing a stripping column was more economical than using a conventional distillation column for both the binary and ternary systems. In the ternary systems with two recycle streams, when the product was the intermediate component in terms of boiling point, the utilization of a divided-wall distillation column (DWC) resulted in a total annual cost saving of 35 % and 41 % compared to direct and indirect separation methods, respectively.

A Numerical Integrator for Kinetostatic Folding of Protein Molecules Modeled as Robots with Hyper Degrees of Freedom

The kinetostatic compliance method (KCM) models protein molecules as nanomechanisms consisting of numerous rigid peptide plane linkages. These linkages articulate with respect to each other through changes in the molecule dihedral angles, resulting in a kinematic mechanism with hyper degrees of freedom. Within the KCM framework, nonlinear interatomic forces drive protein folding by guiding the molecule’s dihedral angle vector towards its lowest energy state in a kinetostatic manner. This paper proposes a numerical integrator that is well suited to KCM-based protein folding and overcomes the limitations of traditional explicit Euler methods with fixed step size. Our proposed integration scheme is based on pseudo-transient continuation with an adaptive step size updating rule that can efficiently compute protein folding pathways, namely, the transient three-dimensional configurations of protein molecules during folding. Numerical simulations utilizing the KCM approach on protein backbones confirm the effectiveness of the proposed integrator.

CFD Simulations and Phenomenological Modelling of Aerodynamic Stall Hysteresis of NACA 0018 Wing

Computational simulations of three-dimensional flow around a NACA 0018 wing with an aspect ratio of AR=5 were carried out by using the Unsteady Reynolds-Averaged Navier–Stokes (URANS) equations with the Shear-Stress Transport turbulence model closure. Simulations were performed to capture aerodynamic stall hysteresis by using the developed pseudo-transient continuation (PTC) method based on a dual-time step approach in CFD OpenFOAM code. The flow was characterized by incompressible Mach number M=0.12 and moderate Reynolds number Re=0.67×106. The results obtained indicate the presence of noticeable aerodynamic hysteresis in the static dependencies of the force and moment coefficients, as well as the manifestation of bi-stable flow separation patterns, accompanied by the development of asymmetry in the stall zone. The URANS simulation results are in good agreement with the experimental data obtained for the NACA 0018 finite-aspect-ratio wing in the low-speed wind tunnel under the same test conditions. A new phenomenological bifurcation model of aerodynamic stall hysteresis under static and dynamic conditions is formulated and is proven to be able to closely match the experimental data.

A pseudo transient nonequilibrium method for rigorous simulation of multicomponent separation columns

Nonequilibrium stage model is a significant improvement in multicomponent separation process simulation, but more equations are involved and the solution of the model equations, which relies on an adequate initial guess for convergence of the Newton method, is difficult. In this work, based on the concept of pseudo-transient continuation approach, we proposed a new pseudo-transient (PT) nonequilibrium method. The proposed method decouples the strongly coupled model equations by introducing dynamic equations for material and energy conservation, as well as transition equations. Thus, the steady-state solution of the nonequilibrium stage model can be obtained through a robust and fast integration process, and the initial guess issue in Newton method can be effectively avoided. Two simulation cases were used to demonstrate the advantages and applicability of the proposed PT nonequilibrium method.

Steady-state density-driven flow and transport: Pseudo-transient parameter continuation

Density-driven flow and transport is one of the dominant physical processes in coastal aquifer systems. To better manage water resources in coastal areas, the interactions need to be understood between freshwater and denser seawater. Numerical models often suffer from excessive computational cost to simulate the equilibrium steady state due to the nonlinear nature of the governing equations to describe density-driven flow. This study suggests a pseudo-transient parameter continuation (PTPC) approach which combines two general solution strategies for nonlinear steady-state equations: the pseudo-transient continuation (PTC) and parameter continuation (PC) methods. The new method gradually adjusts the maximum density (or parameter continuation) in the transient simulation framework (or pseudo-transient). The PTPC method is generally supposed to converge as in PTC and as efficiently as PC when the scheduling for parameter adjustment is set to be optimal. Through a proof-of-concept example, it is illustrated that the solution path to the steady state in PTPC is in between the paths from the PTC and PC approaches, depending on the level of aggressiveness of the parameter adjustment scheduling. As yet being empirical, the PTPC approach is promising and can be applicable to efficiently solve other steady-state nonlinear equations. The method was successfully applied to characterize and better understand the groundwater flow system in the Suruga Bay area, Japan.

Simulation and optimization of VPSA system based on pseudo transient continuation method

AbstractIn order to address the challenge of optimization of pressure swing adsorption systems, an optimization framework based on pseudo transient continuation method was used and a corresponding library with several models was established for simulation of more complex processes. Then, two vacuum pressure swing adsorption processes for oxygen and nitrogen production were considered as case studies and the results based on the single discretization and pseudo transient method were compared and analyzed in detail. Besides, a new strategy was proposed for the stability and efficiency of simulation. Finally, reduced successive quadratic programming and time relaxation algorithm were used for the optimization of the two systems, respectively, and the results showed that the time relaxation algorithm based on the pseudo transient framework has a strong advantage in saving optimization time cost compared with the former.

Explicit pseudo-transient continuation and the trust-region updating strategy for unconstrained optimization

This paper considers an explicit continuation method and the trust-region updating strategy for the unconstrained optimization problem. Moreover, in order to improve its computational efficiency and robustness, the new method uses the switching preconditioning technique. In the well-conditioned phase, the new method uses the L-BFGS method as the preconditioning technique in order to improve its computational efficiency. Otherwise, the new method uses the inverse of the Hessian matrix as the pre-conditioner in order to improve its robustness. Numerical results also show that the new method is more robust and faster than the traditional optimization method such as the trust-region method and the line search method. The computational time of the new method is about one percent of that of the trust-region method (the subroutine fminunc.m of the MATLAB2019a environment is set by the trust-region method) or one fifth of that of the line search method (fminunc.m is set by the quasi-Newton method) for the large-scale problem. Finally, the global convergence analysis of the new method is also given.

A residual smoothing strategy for accelerating Newton method continuation

A technique for accelerating global convergence of pseudo-transient continuation Newton methods is proposed based on residual smoothing. The technique is motivated by the effectiveness of local nonlinear smoothers at overcoming strong nonlinear transients. In the limit of a small pseudo-time step, the method reduces to a local nonlinear smoothing technique, while in the limit of large pseudo-time steps, an exact Newton method is recovered along with its quadratic convergence properties. The formulation relies on the addition of a smoothing source term while leaving the Newton Jacobian matrix unchanged, thus simplifying implementations for existing Newton solvers. The proposed technique is demonstrated on a steady-state and an implicit time-dependent computational fluid dynamics problem, showing significant gains in overall solution efficiency.

A fast ADI algorithm for nonlinear Poisson equation in heterogeneous dielectric media

Recently, a nonlinear Poisson equation has been introduced to model nonlinear and nonlocal hyperpolarization effects in electrostatic solute-solvent interaction for biomolecular solvation analysis. Due to a strong nonlinearity associated with the heterogeneous dielectric media, this Poisson model is difficult to solve numerically, particularly for large protein systems. A new pseudo-transient continuation approach is proposed in this paper to efficiently and stably solve the nonlinear Poisson equation. A Douglas type alternating direction implicit (ADI) method is developed for solving the pseudo-time dependent Poisson equation. Different approximations to the dielectric profile in heterogeneous media are considered in the standard finite difference discretization. The proposed ADI scheme is validated by considering benchmark examples with exact solutions and by solvation analysis of real biomolecules with various sizes. Numerical results are in good agreement with the theoretical prediction, experimental measurements, and those obtained from the boundary value problem approach. Since the time stability of the proposed ADI scheme can be maintained even using very large time increments, it is efficient for electrostatic analysis involving hyperpolarization effects.

Optimal design of extractive dividing-wall column using an efficient equation-oriented approach

The extractive dividing-wall column (EDWC) is one of the most efficient technologies for separation of azeotropic or close boiling-point mixtures, but its design is fairly challenging. In this paper we extend the hybrid feasible path optimisation algorithm (Ma Y, McLaughlan M, Zhang N, Li J. Computers & Chemical Engineering, 2020, 143: 107058) for such optimal design. The tolerances-relaxation integration method is refined to allow for long enough integration time that can ensure the solution of the pseudo-transient continuation simulation close to the steady state before the required tolerance is used. To ensure the gradient and Jacobian information available for optimisation, we allow a relaxed tolerance for the simulation in the sensitivity analysis mode when the simulation diverges under small tolerance. In addition, valid lower bounds on purity of the recycled entrainer and the vapour flow rate in column sections are imposed to improve computational efficiency. The computational results demonstrate that the extended hybrid algorithm can achieve better design of the EDWC compared to those in literature. The energy consumption can be reduced by more than 20% compared with existing literature report. In addition, the optimal design of the heat pump assisted EDWC is achieved using the improved hybrid algorithm for the first time.

Smooth Local Correlation-Based Transition Model for the Spalart–Allmaras Turbulence Model

A smooth version of the local correlation-based transition model (LM2015) for the Spalart–Allmaras (SA) turbulence model is presented. The LM2015 model with helicity-based crossflow correlations is modified and coupled to the SA turbulence model, designated SA-LM2015. The LM2015 and SA-LM2015 transition models include source terms that contain stiff and nonsmooth functions. Approximations to these functions are introduced to eliminate discontinuities and improve the numerical behavior of the model, with the smooth model designated SA-sLM2015. Deep convergence is achieved using a fully coupled, implicit Newton–Krylov algorithm globalized using an efficient pseudotransient continuation strategy. Modifications to the Newton–Krylov algorithm are introduced, including a source-term time stepping strategy, to address the large sources introduced by the turbulence and transition model equations. Two- and three-dimensional transition test cases demonstrate that both models, SA-LM2015 and SA-sLM2015, are able to predict transition due to a variety of mechanisms accurately, with the smooth variant displaying significantly improved numerical behavior. Several of the strategies presented, including the smoothing techniques and source-term time stepping, could be useful in the context of other transition models as well.

Novel feasible path optimisation algorithms using steady-state and/or pseudo-transient simulations

The feasible path optimisation algorithm is widely used for chemical process optimisation due to its effectiveness for large-sale highly nonlinear optimisation problems. The effectiveness strongly relies on the convergence of process simulation as the simulation is performed in each optimisation step. Existing feasible path optimisation algorithms often fail to converge or require expensive computational effort in process simulation in the equation-orientated environment. In this work, we propose three novel feasible path optimisation algorithms to improve both convergence and computational efficiency. The first algorithm improves the original steady-state feasible path algorithm through resetting the initial point used for process simulation when its direct precedence fails during the line search. The second algorithm is an enhancement to the time-relaxation-based optimisation algorithm through the use of the tolerances-relaxation integration method for the pseudo-transient continuation (PTC) simulation. The last algorithm is a hybrid algorithm through the effective combination of steady-state simulation and PTC simulation in the feasible path optimisation framework. The computational results demonstrate that the proposed three new variants do improve the convergence and are more robust than the existing feasible path optimisation algorithms. The first variant is more efficient than the other two and the existing PTC model-based optimisation algorithms with the same computer hardware and software.

A dynamically load-balanced parallel p-adaptive implicit high-order flux reconstruction method for under-resolved turbulence simulation

We present a dynamically load-balanced parallel p-adaptive implicit high-order flux reconstruction method for under-resolved turbulence simulation. The high-order explicit first stage, singly diagonal implicit Runge–Kutta (ESDIRK) method is employed to circumvent the restriction on the time step size. The pseudo transient continuation is coupled with the matrix-free restarted generalized minimal residual (GMRES) method to solve the nonlinear equations at each stage, except the first one, of ESDIRK. We use the spectral decay smoothness indicator as the refinement/coarsening indicator for p-adaptation. A dynamic load balancing technique is developed with the aid of the open-source library ParMETIS. The trivial cost, compared to implicit time stepping, of mesh repartitioning and data redistribution enables us to conduct p-adaptation and load balancing every time step. An isentropic vortex propagation case is employed to study the impact of element weights used in mesh repartitioning on parallel efficiency. We apply the p-adaptive solver for implicit large eddy simulation (ILES) of the transitional flows over a cylinder when Reynolds number (Re) is 3900 and the SD7003 wing when Re is 60000. Numerical experiments demonstrate that a significant reduction in the run time (up to 70%) and total number of solution points (up to 76%) can be achieved with p-adaptation.

Comparison of ROW, ESDIRK, and BDF2 for Unsteady Flows with the High-Order Flux Reconstruction Formulation

We conduct a comparative study of the Jacobian-free linearly implicit Rosenbrock–Wanner (ROW) methods, the explicit first stage, singly diagonally implicit Runge–Kutta (ESDIRK) methods, and the second-order backward differentiation formula (BDF2) for the high-order flux reconstruction/correction procedure via reconstruction solution of the unsteady Navier–Stokes equations. Pseudo-transient continuation is employed to solve the nonlinear equation at each stage of ESDIRK (excluding the first stage) and each step of BDF2. A Jacobian-free implementation of the restarted generalized minimal residual method solver is employed with a low storage element-Jacobi preconditioner to solve the linear system at each stage of ROW and each pseudo time iteration of ESDIRK and BDF2. Several numerical experiments, including both laminar and turbulent flow simulations, are conducted to carry out the comparison. We observe that the multistage ROW2 and ESDIRK2 are more efficient than the multistep BDF2, and higher-order implicit time integrators are more efficient than lower-order ones. In general, the ESDIRK method allows a larger physical time step size for unsteady flow simulation than the ROW method when the element-Jacobi preconditioner is employed, especially for wall-bounded flows; and the ROW method can be more efficient than the ESDIRK method when the time step size is refined.

Efficient Convergence for a Higher-Order Unstructured Finite Volume Solver for Compressible Flows

This Paper presents a three-dimensional higher-order-accurate finite volume algorithm for the solution of steady-state compressible flow problems. Higher-order accuracy is achieved by constructing a piecewise continuous representation of the average solution values using the -exact reconstruction scheme. The pseudo-transient continuation method is employed to reduce the solution of the discretized system of nonlinear equations into the solution of a series of linear systems, which are subsequently solved using the generalized minimal residual (GMRES) method. This Paper considers several preconditioning methods in conjunction with different matrix reordering algorithms and shows that the proposed preconditioner based on inner GMRES iterations can enhance the convergence speed and reduce the memory cost of the solver. Moreover, when starting from a lower-order solution as the initial condition, this Paper shows that ramping up the Courant–Friedrichs–Lewy (CFL) number accelerates the convergence rate. Finally, this Paper verifies the developed finite volume algorithm by solving a set of test problems, in which optimal solution convergence with mesh refinement is attained.

A Dynamic Theory-Based Method for Computing Unstable Equilibrium Points of Power Systems

In this paper, a new theory-based dynamic method for computing unstable equilibrium points is proposed. This dynamic method is a combination of a dynamic transformation method and a trajectory-unified method for the computation of unstable equilibrium points (UEPs) of power system models. The transformation method converts a UEP into a stable equilibrium point (SEP) to expand its convergence region by creating a quotient gradient system (QGS). The resulting SEP is then calculated using a quasi–Newton form of the pseudo-transient continuation method ( ${\boldsymbol \psi }\boldsymbol{tc}$ ) that exploits the structure of the quotient gradient system for fast and reliable computation. It is shown that the proposed QGS-based ${\boldsymbol \psi } \boldsymbol{tc}$ can have local q -superlinear or even local quadratic convergence under certain conditions. These conditions for convergence are presented and analyzed. The proposed method is tested on the WSCC 9-bus system and the IEEE 145-bus 50-machine system. The results show that the proposed method gives accurate results, is sufficiently fast, numerically stable, and enlarges the convergence region of computed UEPs.

A Jacobian-free approximate Newton–Krylov startup strategy for RANS simulations

The favorable convergence rates of Newton–Krylov-based solution algorithms have increased their popularity for computational fluid dynamics applications. Unfortunately, these methods perform poorly during the initial stages of convergence, particularly for three-dimensional Reynolds-averaged Navier–Stokes simulations. Addressing this problem requires the use of a globalization method such as pseudo-transient continuation, along with an approximate Newton–Krylov startup stage. This class of methods marches the solution in pseudo-time with a matrix-based approximate Jacobian that has a lower bandwidth and better conditioning properties compared with the exact Jacobian. However, this matrix-based approach also has shortcomings, including the large cost of computing and storing the approximate Jacobian and its preconditioner, along with the need for an updated Jacobian for every nonlinear iteration. To rectify these shortcomings, we use approximate residual formulations in a Jacobian-free approximate Newton–Krylov algorithm. With the approximate Jacobian, we compute the vector products by using the approximate residual computations in a matrix-free manner while forming a preconditioner based on the matrix-based approximate Jacobian. This approach keeps the approximate Jacobian up to date and mitigates the cost of forming a matrix-based Jacobian and its preconditioner at each iteration by lagging the preconditioner between nonlinear iterations. We use varying levels of approximations with the matrix-free approach and thereby demonstrate the trade-off between rate of convergence and the cost of each nonlinear iteration. The proposed implementation uses only the exact and approximate residual formulations and can therefore be generalized with minimal additional implementation effort to a range of solvers and discretizations. The code is available under an open-source license.

Rate-based modeling and economic optimization of next-generation amine-based carbon capture plants

Amine scrubbing processes remain an important technology for mitigating the contribution of carbon emissions to global warming and climate change. Like other chemical processes, they can benefit from computer-aided optimization at the design stage, but systematic optimization procedures are rarely employed due to the challenges of simulating the requisite rate-based mass transfer and reaction models. This paper presents a novel approach for the simulation and optimization of rate-based columns, with specific application to the absorber and stripper columns found in (amine-) solvent-based carbon capture processes. The approach is based on pseudo-transient continuation, and the resulting column models are easily incorporated into large-scale process flowsheets with other previously developed pseudo-transient models. We demonstrate that the proposed approach allows for gradient-based optimization of next-generation amine scrubbing processes by considering a complex carbon capture process under three different operating conditions. The results provide general insight into the design of amine scrubbing processes, and shadow prices at the optimal point(s) suggest potential avenues for improving the process economics. The effects of carbon dioxide removal percentage and flue gas composition on process economics are briefly analyzed.

Equation-oriented optimization of methanol distillation systems using pseudo-transient models

Methanol distillation is an energy-intensive and complex integrated process. Up to now, methanol distillation schemes involving energy-saving technologies have been developed based on process simulation, or based on optimization under certain simplified conditions because of the problem of convergence and integer optimization. In this work, six distillation schemes, two of which are proposed for the first time, were optimized using a steady-state optimization algorithm assisted by rigorous pseudo-transient continuation (PTC) models in equation-oriented environments. The optimization results indicated that double-effect distillation could significantly reduce the energy consumption, where the implementation of heat integration had an evident impact on both the energy cost and capital cost. The economic analysis showed that, under the same feed conditions, the total annual cost (TAC) of the five-column scheme was the lowest. There are three groups of heat integrations in the scheme. Since the energy consumption of the methanol distillation process accounts for a large proportion in TAC, adopting multi-effect distillation technology is a good way to save energy in this situation. Although at the expense of increasing the capital cost, the five-column scheme is still the most economic owing to its significant energy saving effect. Finally, after overall comparison of the six schemes, the optimal scheme characteristics are summarized, which is of great reference value for the future design of methanol distillation systems.

Towards the really optimal design of distillation systems: Simultaneous pressures optimization of distillation systems based on rigorous models

Operating pressures have a significant impact on the economics of distillation systems, but most works optimizing distillation systems using rigorous models neglected the determination of optimal operating pressures. Thus, in this work, we will show the influence of the pressure on mathematical optimization-based design using rigorous distillation models. For the optimization, the pseudo-transient continuation model is used to guarantee the robustness, and the bypass efficiency method is used to optimize the number of stages efficiently. Furthermore, to obtain accurate integer solutions reliably, an improved successive rounding method is proposed to combine with the steady-state optimization algorithm assisted by PTC models. Finally, the optimization results of several intensified/integrated distillation systems persuasively presented that the operating pressures did have a significant influence on optimization-based designs. Also, the rigorous model-based optimization algorithms got much better designs than both heuristic-based method and short-cut model-based optimization algorithm (with more than 10% decreases in TAC).