An augmented Lagrangian interior-point approach for large-scale NLP problems on graphics processing units

The demand for fast solution of nonlinear optimization problems, coupled with the emergence of new concurrent computing architectures, drives the need for parallel algorithms to solve challenging nonlinear programming (NLP) problems. In this paper, we propose an augmented Lagrangian interior-point approach for general NLP problems that solves in parallel on a Graphics processing unit (GPU). The algorithm is iterative at three levels. The first level replaces the original problem by a sequence of bound-constrained optimization problems using an augmented Lagrangian method. Each of these bound-constrained problems is solved using a nonlinear interior-point method. Inside the interior-point method, the barrier sub-problems are solved using a variation of Newton's method, where the linear system is solved using a preconditioned conjugate gradient (PCG) method, which is implemented efficiently on a GPU in parallel. This algorithm shows an order of magnitude speedup on several test problems from the COPS test set.     

Simple, robust, and fast iterative solution of Underwood's equation for minimum reflux

This work has developed a simple, robust, and fast method for the solution of Underwood's equation f(x) for minimum reflux. The new scheme involves devising an iterative solver g by re-arranging this equation, obtaining a secondary function equal to g − x, and finally applying Ostrowski's fourth-order technique to find the roots of this function. The use of Ostrowski's method in place of Newton's popular second-order formula requires little extra calculation per iteration other than a function value at an auxiliary point. The novel method is successful where the direct applications of Newton's and Ostrowski's solvers to Underwood's equation fail. It keeps iterations within bounds and rapidly converges to the root of interest.     

Interior-point decomposition approaches for parallel solution of large-scale nonlinear parameter estimation problems

Multi-scenario optimization is a convenient way to formulate and solve multi-set parameter estimation problems that arise from errors-in-variables-measured (EVM) formulations. These large-scale problems lead to nonlinear programs (NLPs) with specialized structure that can be exploited by the NLP solver in order to obtained more efficient solutions. Here we adapt the IPOPT barrier nonlinear programming algorithm to provide efficient parallel solution of multi-scenario problems. The recently developed object oriented framework, IPOPT 3.2, has been specifically designed to allow specialized linear algebra in order to exploit problem specific structure. This study discusses high-level design principles of IPOPT 3.2 and develops a parallel Schur complement decomposition approach for large-scale multi-scenario optimization problems. A large-scale case study example for the identification of an industrial low-density polyethylene (LDPE) reactor model is presented. The effectiveness of the approach is demonstrated through the solution of parameter estimation problems with over 4100 ordinary differential equations, 16,000 algebraic equations and 2100 degrees of freedom in a distributed cluster.     

3D Viscoelastic Simulation of Jetting in Injection Molding

A three‐dimensional model and algorithm were proposed to simulate the jetting phenomenon in injection molding. The Giesekus model was employed to describe the viscoelastic behavior of the incompressible, non‐isothermal polymer flow, and the volume of filling method was used to track the jetting free surface and advancing melt front. The governing and constitutive equations were discretized by finite volume method. To improve the precision and stability of the numerical method, a modified pressure implicit with splitting of operator algorithm was proposed to solve the pseudo Navier–Stokes problem, and an iterative scheme was constructed to determine the solution of the coupled nonlinear equations. Numerical results showed this method successfully predicted the snake‐like flow in thin‐wall cavities and buckling flow in thick‐wall cavities. The viscoelastic characteristics of polymer influenced the reptile swing frequency and amplitude. Finally, a suitable nonlinear parameter in the Giesekus model improved the precision of simulation. POLYM. ENG. SCI., 59:E397–E405, 2019. © 2019 Society of Plastics Engineers     

Incorporating Darcy's law for pure solvent flow through porous tubes: Asymptotic solution and numerical simulations

A generalized solution for pressure‐driven, incompressible, Newtonian flow in a porous tubular membrane is challenging due to the coupling between the transmembrane pressure and velocity. To date, all analytical solutions require simplifications such as neglecting the coupling between the transmembrane pressure and velocity, assuming the form of the velocity fields, or expanding in powers of parameters involving the tube length. Moreover, previous solutions have not been validated with comparison to direct numerical simulation (DNS). We comprehensively revisit the problem to present a robust analytical solution incorporating Darcy's law on the membrane. We make no assumptions about the tube length or form of the velocity fields. The analytic solution is validated with detailed comparison to DNSs, including cases of axial flow exhaustion and cross flow reversal. We explore the validity of typical assumptions used in modeling porous tube flow and present a solution for porous channels in Supporting Information. © 2012 American Institute of Chemical Engineers AIChE J, 2012     

A solution method for solving coupled nonlinear boundary-value problems

A Strum-Liouville integral transform technique is novelly applied to solve system of coupled nonlinear boundary-value problems approximately. The systems of differential equations consist of a linear differential operator and a nonlinear function of the dependent variables. To illustrate the potential of this technique we consider an example which comes from the modeling of diffusion and nonlinear chemical reaction systems in chemical engineering. The approximate solutions obtained by our technique agree surprising well with the numerically exact solutions obtained by the orthogonal collocation technique. To improve the approximation an iteration scheme in transform space is also defined.

Scope—Today, mathematical modeling of physical phenomena often produces (single or coupled) nonlinear differential equations. The true physical situation can, in many cases, be more closely described if the differential equations are allowed to be nonlinear. However, nonlinear differential equations are generally too difficult to be solved analytically apart from a few “tricks” or substitutions which apply only to a handful of equations [1]. An alternative approach is to look for a method which will reduce the problem, via analytical techniques, to a point where a “simple” computer program can solve the rest of the problem. The method introduced in this paper belongs to this class of solution techniques.

The method, which in this paper is applied to solving coupled nonlinear boundary-value problems, is a generalization of an idea in a paper by Do and Bailey [2] who apply it to a single nonlinear differential equation of boundary-value type. The equations, to which the technique is applied, arise from Fick's law diffusion into a porous solid and nonlinear reaction within the solid.

The solution method employs a Strum-Loiuville integral transform and to account for the nonlinear part an approximation is introduced. An iteration scheme is defined to improved the accuracy of the solution. The system of coupled nonlinear differential equations is reduced to a system of coupled nonlinear algebraic equations which is solved using a Newton-Raphson process. Finally, the solution is expressed as an infinite series, which is summed using a computer.

In response to papers by Do and Bailey [3] and Do and Weiland [4], Jerri [5] has tried to put this method on a more mathematical footing, and he shows that this method is a special case of a more general technique he has devised. Jerri uses the idea of Fourier transforms and convolution products to justify his method. The results for the example he considered are good, but he did not state how many iterations he required to obtain the solutions reported.

Conclusions and Significance—This paper has presented a very powerful method of solving boundary-value problems with linear operators and a nonlinear function of the dependent variable. The method works well for a single equation or coupled equations and can handle any kind of nonlinear function. We have shown through extensive numerical calculation the accuracy of this solution method, where the accuracy is measured in terms of a ratio of norms. In most cases an error of 4% can be achieved with just one iteration (Tables 2 and 3). Even though the present method has been applied to problems which have arisen from the modeling of chemical engineering problems, it would also be applicable to differential equations arising in other areas, provided they are of the same form.     

Geometrical analysis of dynamic problem on the membrane transport using spectral solution

The problem analyzed in this paper is a specific application of the composite membrane. General diffusion and convection formulation is presented for the dynamic problem. The spectral analysis considers convective transport of a single solute species across a one dimensional membrane system. The solution is obtained using operator theoretic methods. The geometrical structure of the spectrum of the operator is determined for the complete range of the various parameters including the distribution coefficient, the convective velocity and the diffusion coefficient. The structure of the spectrum allows a complete characterization of all the eigenvalues of the system in terms of all of these physical parameters. Calculation of the first eigenvalue for a number of cases shows its variation with the convective velocity for various medium porosities and allows a priori estimates of the profiles.     

基于广义原理的对流换热热阻计算方法及检验

使用积耗散与广义热阻原理分析对流换热中耗散与广义换热温差间的关系, 提出了控制体内三种广义热阻值计算方法的表达形式, 并在二维平行通道层流换热中进行检验, 与传统热阻计算方法作比较后发现传统的表面传热系数不能准确体现研究区域内的热导（阻）能力, 三种广义热阻计算方法的表达形式相互吻合, 结果一致, 且对流换热退化为热传导时与定义式和热传导计算结果统一, 表明是可靠的对流热阻计算方法; 使用该方法研究二维平行通道内入口段的对流换热, 发现广义热阻值的普遍趋势是随流动发展热阻增大, 符合对流换热的基本理念, 定义式热阻则表现变化很大, 受温差与壁面热流量影响明显; 检验中发现了广义式热阻与定义式热阻的差异是由于定义式热阻为简化表达导致。     

GENERALIZED FINITE DIFFERENCE SOLUTION FOR HEAT AND MASS TRANSFER FROM A FINITE CYLINDER DURING QUENCH

A generalized nondimensional solution is presented that describes heat or mass transfer from a finite cylinder during quench. The solution is applicable to three important cases:

Conduction with convection heat transfer at the surface during any single step hot or cold quench.

Conduction with radiation heat transfer at the surface during a single step cold quench with negligible background radiation.

Diffusion with surface desorption of a diatomic gas from a metal specimen during a single step quench in a high vacuum with negligible background pressure.

Application of the generalized solution, which utilizes the numerical method of finite differences with forward stepping, is illustrated by determining a cylinder's transient temperature distribution and surface transfer rate (both instantaneous and cumulative) for an example L/D ratio of 2.0. Selected results are graphed and tabulated for the three cases. The results for the conduction/convection case are verified using the familiar analytical product solution as well as the lumped solution. For the conduction/radiation and diffusion/desorption cases, no analytical solutions are available other than the lumped limit which is in agreement.     

Numerical simulation of particle formation in the rapid expansion of supercritical solution process

In this paper, we numerically study particle formation in the rapid expansion of supercritical solution (RESS) process in a two dimensional, axisymmetric geometry, for a benzoic acid + CO₂ system. The fluid is described by the classical Navier–Stokes equation, with the thermodynamic pressure being replaced by a generalized pressure tensor. Homogenous particle nucleation, transport, condensation and coagulation are described by a general dynamic equation, which is solved using the method of moments. The results show that the maximal nucleation rate and number density occurs near the nozzle exit, and particle precipitation inside the nozzle might not be ignored. Particles grow mainly across the shocks. Fluid in the shear layer of the jet shows a relatively low temperature, high nucleation rate, and carries particles with small sizes. On the plate, particles within the jet have smaller average size and higher geometric mean, while particles outside the jet shows a larger average size and a lower geometric mean. Increasing the preexpansion temperature will increase both the average particle size and standard deviation. The preexpansion pressure does not show a monotonic dependency with the average particle size. Increasing the distance between the plate and the nozzle exit might decrease the particle size. For all the cases in this paper, the average particle size on the plate is on the order of tens of nanometers.     

Direct stamping and capillary flow patterning of solution processable piezoelectric polyvinylidene fluoride films

It is well known that the potential applications of polyvinylidene fluoride (PVDF) mainly come from the piezoelectricity and ferroelectricity of its polar β phase. Thus, we have investigated the effect of different preparation conditions namely evaporation temperature, type of solvent and additive to enhance the β crystal structures of PVDF thin film. Subsequently, facile and direct soft lithography technique; direct stamping and capillary flow were employed to demonstrate good pattern transfer of PVDF thin films. The piezoelectricity of the microstructure was characterized using piezoresponse force microscopy (PFM) where fairly good piezoresponse was obtained without further processing procedures i.e., annealing or applied pressure/electric field. As such, our solution processable and direct patterning of PVDF techniques offer facile and promising route to produce arrays of isolated microstructures with improved piezoelectric functionality.     

A flexible solution method for generalized equilibrium stage columns

This paper presents a flexible solution method for the process design and simulation of generalized equilibrium stage absorption or distillation columns for solving a wide range of multistage, multicomponent separation problems in the petroleum or petrochemical industries. Although the model was developed to increase the control flexibility of product components in a solution, it was found that it also increases the efficiency of convergence at the expense of greater core usage. The mathematical model includes overall material balances, component material balances, energy balances, summation equations and specification equations. These nonlinear equations are solved simultaneously via the matrix partitioning technique together with the popular Newton-Raphson iterative algorithm. The method is applicable for both absorption and distillation columns with multiple feeds and sidedraws. This model will offer more flexible choices of the column specifications such as tray temperature, overhead product rate, reflux ratio, boilup ratio, tray vapor/liquid flow and product purity/recovery. The composition dependent equilibrium and enthalpy correlations such as Chao-Seader, Grayson-Streed, and Soave-Redlich-Kwong are incorporated into the mathematical model. Most problems can be easily converged with less than ten iterations.     

Numerical solution approaches for robust nonlinear optimal control problems

Nonlinear equality and inequality constrained optimization problems with uncertain parameters can be addressed by a robust worst-case formulation that leads to a bi-level min–max optimization problem. We propose and investigate a numerical method to solve this min–max optimization problem exactly in the case that the underlying maximization problem always has its solution on the boundary of the uncertainty set. This is an adoption of the local reduction approach used to solve generalized semi-infinite programs. The approach formulates an equilibrium constraint employing first order derivatives of both the uncertainty set and the user defined constraints. We propose two different ways for computation of these derivatives, one similar to the forward mode, the other similar to the reverse mode of automatic differentiation. We show the equivalence of the proposed approach to a method based on geometric considerations that was recently developed by some of the authors. We show how to generalize the techniques to optimal control problems. The robust dynamic optimization of a batch distillation illustrates that both techniques are numerically efficient and able to overcome the inexactness of another recently proposed numerical approach to address uncertainty in optimal control problems.     

NOTE ON A TWO-PHASE ASYMMETRIC STROKES FLOW

The solution of Stokes equations for a rotating axisymmetric body which straddles an interface between two immiscible fluids is difficult to obtain in the absence of symmetry about the planar interface. This note presents a method of dealing with slightly asymmetric problems.     

NOTE ON A TWO-PHASE ASYMMETRIC STROKES FLOW

The solution of Stokes equations for a rotating axisymmetric body which straddles an interface between two immiscible fluids is difficult to obtain in the absence of symmetry about the planar interface. This note presents a method of dealing with slightly asymmetric problems.     

Optimization of parallel cyclones

A new procedure is presented to optimize the number and size of parallel cyclones. For a given efficiency, constraints due to inlet velocity and pressure drop limitations are taken into account.An analytical solution is obtained, and an iterative method of calculation is proposed to check constraints and to search for the existence of a ‘better’ optimum at lower values of d₁. The method can also be applied to optimize the size of a single cyclone.     

Numerical simulation of three‐dimensional gas‐solid particle flow in a horizontal pipe

A three‐dimensional model of particulate flows using the Reynolds Averaged Navier‐Stokes method is presented. The governing equations of the gas–solids flow are supplemented with appropriate closure equations to take into account all the relevant forces exerted on the solid particles, such as particle‐turbulence interactions, turbulence modulation, particle–particle interactions, particle–wall interactions, as well as gravitational, viscous drag, and lift forces. A finite volume numerical technique was implemented for the numerical solution of the problem. The method has been validated by comparing its results with the limited number of available experimental data for the velocity and turbulence intensity of the gas–particle flow. The results show that the presence of particles in the flow has a significant effect on all the flow variables. Most notably, the distribution of all the parameters becomes asymmetric, because of the gravitational effect on the particles and particle sedimentation. © 2011 American Institute of Chemical Engineers AIChE J, 2011     

Preparation of defect-free asymmetric membranes for gas separations

A technique was developed to prepare defect-free, asymmetric, polymer membranes for gas separation. The preparation method eliminates the need for coatings, which are usually required to render asymmetric, polymer based, membranes gas selective. In this method, a casting solution containing a polymer, solvent, and salt additive is given a desired shape and immersed in a coagulation bath containing a nonsolvent. The nonsolvent is selected to have a low affinity for both the solvent and salt additive. After the complete coagulation of the membrane, the additive salt is leached out in a second bath. This leads to the formation of an asymmetric membrane that has a well-interconnected porous network. The fine membrane structure is preserved by solvent exchange before it is finally dried. Polyetherimide (PEI) (Ultem® 1000) membranes were prepared from casting solutions containing 23, 25, and 26.5% (wt) PEI, various amounts of lithium nitrate and N-methyl-2-pyrrolidinone (NMP). Membrane performance was determined for the separation of oxygen from air. The effects of polymer concentration, additive salt concentration and the drying process on oxygen permeance, and the actual separation factor of the membrane are discussed. The addition of a small amount of solvent to the coagulation bath improved the leaching of the salt additive and produced membranes with a more open structure. A polymer concentration of 23% produced membranes with the highest performance. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 73: 1471–1482, 1999     

Mixed convection mass transfer studies of opposing and aiding flow in a parallel plate electrochemical flow cell

Studies of combined natural and forced convection in a vertical parallel plate electrochemical cell in laminar conditions in cases of opposing and aiding flow are reported. In an ongoing project it was necessary to identify conditions in which natural convection had no significant influence on mass transfer rates at the cell walls so that data could be validly compared with purely laminar flow computational models. For the different electrode lengths investigated, natural convection dominated at low Reynolds number and there was no Reynolds number dependence. At high Reynolds number the data approached the laminar flow solution. At intermediate Reynolds number, however, there existed a distinct region where free and forced convection were significant. At high electrolyte concentrations data did not merge with laminar flow equations until Re=1000 and low electrolyte concentration data for the large plate could not be compared with numerical predictions below Re of 250. An attempt was made to compare the data with those of other workers on combined forced and natural convection heat and mass transfer.     

CFD analysis of flow field in square cyclones

In this study, computational fluid dynamic method is used to predict and evaluate the flow field inside a square cyclone. The flow field is calculated using 3D Reynolds-averaged Naveir-Stokes equations. The Reynolds stress transport model (RSTM) is used to simulate the Reynolds stresses. The Eulerian-Lagrangian computational procedure is implemented to predict particle trajectory in the cyclone. The Newton's second law is used to study the particle trajectory with modeling the drag and gravity forces acting on the particles. The velocity fluctuations are simulated using the discrete random walk (DRW). Two square cyclones which have different geometries are studied. The cyclones are simulated at different flow rates. The details of the flow field are studied in the cyclones and the effect of varying the flow rates is observed. Tangential velocity is investigated in different sections inside the square cyclone. Contour of pressure and turbulence intensity is shown for different inlet velocities inside the cyclones. It is observed that different geometries, also different inlet velocities, could affect on the pressure drop. The collection efficiency and the flow patterns obtained numerically are compared with the experimental data and good agreement is observed.