Reliable mixture critical point computation using polynomial homotopy continuation

Sidky, Hythem; Whitmer, Jonathan K.; Mehta, Dhagash

doi:10.1002/aic.15319

Cited by 5 publications

(6 citation statements)

References 53 publications

(106 reference statements)

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“…A huge quantity of numerical procedures can be applied in the solution of this kind of system. Among these, we can mention Newton-type methods (Dimitrakopoulos et al, 2014), homotopy-continuation procedures (Sidky et al, 2016;Wang et al, 1999) and interval methods (Stradi et al, 2001). Here, we employ the methodology of numerical inversion of functions from the plane to the plane, proposed by Malta et al (1993).…”

Section: Numerical Methodologymentioning

confidence: 99%

“…Their algorithm exhibits mathematical and computational guarantees to obtain all critical points. More recently, Sidky, Whitmer, and Mehta (2016) employed a polynomial homotopy continuation algorithm-also a robust method-in the calculation of critical points. Previously, Wang, Wong, Chen, Yan, and Guo (1999) used a homotopy-based method to obtain the critical loci of binary mixtures.…”

Section: Introductionmentioning

confidence: 99%

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Critical Point Calculations by Numerical Inversion of Functions

Parajara,

Platt,

Neto

et al. 2020

Preprint

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In this work, we propose a new approach to the problem of critical point calculation, based on the formulation of Heidemann and Khalil (1980). This leads to a 2 × 2 system of nonlinear algebraic equations in temperature and molar volume, which makes possible the prediction of critical points of the mixture through an adaptation of the technique of inversion of functions from the plane to the plane, proposed by Malta, Saldanha, and Tomei (1993). The results are compared to those obtained by three methodologies: (i) the classical method of Heidemann and Khalil (1980), which uses a double-loop structure, also in terms of temperature and molar volume;(ii) the algorithm of Dimitrakopoulos, Jia, and Li (2014), which employs a damped Newton algorithm and (iii) the methodology proposed by Nichita and Gomez (2010), based on a stochastic algorithm. The proposed methodology proves to be robust and accurate in the prediction of critical points, as well as provides a global view of the nonlinear problem.

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Section: Numerical Methodologymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Critical Point Calculations by Numerical Inversion of Functions

Parajara,

Platt,

Neto

et al. 2020

Preprint

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show abstract

“…Polynomial homotopy continuation has also been adapted to the field of chemical engineering to locate critical points of multicomponent mixtures [SWM16], i.e., temperature and pressure satisfying a multiphase equilibrium.…”

Section: Critical Point Computationmentioning

confidence: 99%

Solving Polynomial Systems with phcpy

Otto¹,

Forbes²,

Verschelde³

2019

Proceedings of the Python in Science Conference

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The solutions of a system of polynomials in several variables are often needed, e.g.: in the design of mechanical systems, and in phase-space analyses of nonlinear biological dynamics. Reliable, accurate, and comprehensive numerical solutions are available through PHCpack, a FOSS package for solving polynomial systems with homotopy continuation.This paper explores new developments in phcpy, a scripting interface for PHCpack, over the past five years. For instance, phcpy is now available online through a JupyterHub server featuring Python2, Python3, and SageMath kernels. As small systems are solved in real-time by phcpy, they are suitable for interactive exploration through the notebook interface. Meanwhile, phcpy supports GPU parallelization, improving the speed and quality of solutions to much larger polynomial systems. From various model design and analysis problems in STEM, certain classes of polynomial system frequently arise, to which phcpy is well-suited.

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“…Finding all solutions can be guaranteed in some cases, for example, continuous polynomial systems. 10 However, constructing a proper homotopy function is very important but it is an experienced task. The global terrain method is another method for finding all mathematical solutions.…”

Section: Introductionmentioning

confidence: 99%

“…This is a sophisticated global method for gradually approaching a solution set for the equations from a starting point that satisfies another simpler system of equations. , In practice, homotopy continuation methods are frequently used in finding all solutions of arbitrary nonlinear systems of equations. Finding all solutions can be guaranteed in some cases, for example, continuous polynomial systems . However, constructing a proper homotopy function is very important but it is an experienced task.…”

Section: Introductionmentioning

confidence: 99%

A Novel Method To Find All Physical Solutions of Constrained Chemical Engineering Models in Polynomial Equations

Zhao

Chen

Zhu

2018

Ind. Eng. Chem. Res.

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Sometimes, there are more than one solutions of a chemical process simulation. In this work, a novel strategy is proposed for finding all physical solutions of chemical engineering models described in polynomials with prescribed variable bounds. First, the Gröbner basis method is proposed to transform the original model into a lower-triangular polynomial system. Based on this triangular structure, the problem of finding solutions of a constrained multivariable polynomial system is converted to solve a set of constrained univariate polynomials sequentially. Next, a real-root-isolation algorithm is developed to compute the disjoint intervals of each univariate polynomial such that each interval contains one solution. Finally, a numerical method with convergence monitoring is proposed to locate each solution. Case studies show that the proposed method can accurately analyze the multisolution features of the systems. All physical solutions can be found directly and effectively.

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Reliable mixture critical point computation using polynomial homotopy continuation

Cited by 5 publications

References 53 publications

Critical Point Calculations by Numerical Inversion of Functions

Critical Point Calculations by Numerical Inversion of Functions

Solving Polynomial Systems with phcpy

A Novel Method To Find All Physical Solutions of Constrained Chemical Engineering Models in Polynomial Equations

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