hp-adaptive least squares spectral element method for population balance equations

Dorao, Carlos Alberto; Jakobsen, Hugo A.

doi:10.1016/j.apnum.2006.12.005

Cited by 27 publications

(16 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In a number of publications Dorao and Jakobsen [8,[30][31][32][33], and Zhu et al [34] investigated the least-squares method for the solution of relatively simple PB problems in which an analytical solution was available in most of the numerical test cases. The least-squares method has been applied to more comprehensive PB problems, e.g.…”

Section: A N U S C R I P Tmentioning

confidence: 99%

Spectral solution of the breakage–coalescence population balance equation Picard and Newton iteration methods

Solsvik

Jakobsen

2016

Applied Mathematical Modelling

View full text Add to dashboard Cite

The behavior of dispersed systems such as gas-liquids or liquid-liquid systems depends on the characteristics of the dispersed phase. The population balance (PB) equation is encountered in numerous engineering disciplines in order to describe complex processes where the accurate prediction of the dispersed phase plays a major role for the overall behavior of the system. In the present study, the orthogonal collocation, Galerkin and least-squares methods have been adopted to solve a non-linear PB equation which consists of both breakage and coalescence terms. The performance of the methods is demonstrated by comparing the numerical solution results with the (manufactured) analytical solution of the problem. For the least-squares method, the choice of linearization technique influences the numerical performance, whereas the Galerkin and orthogonal collocation methods obtain the same numerical accuracy for both the Picard and Newton iteration techniques. The least-squares method suffers from lack of convergency using the Picard method. On the other hand, if the Newton method is employed, the least-squares method obtains the same accuracy as the Galerkin and orthogonal collocation methods.

show abstract

Section: A N U S C R I P Tmentioning

confidence: 99%

Spectral solution of the breakage–coalescence population balance equation Picard and Newton iteration methods

Solsvik

Jakobsen

2016

Applied Mathematical Modelling

View full text Add to dashboard Cite

show abstract

“…Ref. 18 showed the use of an hp−adaptive LS-SEM for solving the population balance equation, while Ref. 19 developed an hp−adaptive spectral element solver for reactor modeling.…”

Section: Introductionmentioning

confidence: 99%

An Improved Adaptive hp Algorithm using Pseudospectral Methods for Optimal Control

Darby

Hager

Rao

2010

AIAA/AAS Astrodynamics Specialist Conference

View full text Add to dashboard Cite

An iterative variable-order hp−adaptive pseudospectral method is presented for solving optimal control problems. In the method of this paper, the accuracy of the solution in each segment is assessed by computing the violations in the discretized differential-algebraic equations at a large number of sample points in each segment. The accuracy on the subsequent mesh is then improved by redistributing the collocation points using a functional that depends upon the curvature of the trajectory. The algorithm iterates on the collocation point distribution until the accuracy criterion is satisfied to a user-specified tolerance. The method allows for different degree polynomial approximations in each segment, but the algorithm is designed in such a manner that the maximum polynomial degree in every segment remains small. It is found that a computationally efficient algorithm results from the ability to vary the degree of the polynomial in each segment while keeping the maximum degree bounded. Finally, an example is considered to demonstrate the effectiveness of the method.

show abstract

“…Reference 26 developed an adaptive spectral least‐squares collocation scheme for the Burgers equation. Reference 27 showed the use of an hp ‐adaptive LS‐SEM for solving the population balance equation whereas Reference 28 developed an hp ‐adaptive spectral element solver for reactor modeling. Finally, an overview of hp ‐adaptive spectral element methods for solving problems in computational fluid dynamics can be found in Reference 29.…”

Section: Introductionmentioning

confidence: 99%

An hp‐adaptive pseudospectral method for solving optimal control problems

Darby¹,

Hager²,

Rao³

2010

Optim Control Appl Methods

404

304

View full text Add to dashboard Cite

An hp-adaptive pseudospectral method is presented for numerically solving optimal control problems. The method presented in this paper iteratively determines the number of segments, the width of each segment, and the polynomial degree required in each segment in order to obtain a solution to a userspecified accuracy. Starting with a global pseudospectral approximation for the state, on each iteration the method determines locations for the segment breaks and the polynomial degree in each segment for use on the next iteration. The number of segments and the degree of the polynomial on each segment continue to be updated until a user-specified tolerance is met. The terminology 'hp' is used because the segment widths (denoted h) and the polynomial degree (denoted p) in each segment are determined simultaneously. It is found that the method developed in this paper leads to higher accuracy solutions with less computational effort and memory than is required in a global pseudospectral method. Consequently, the method makes it possible to solve complex optimal control problems using pseudospectral methods in cases where a global pseudospectral method would be computationally intractable. Finally, the utility of the method is demonstrated on a variety of problems of varying complexity. Determining locations of new segments or increase in number of collocation pointsLet be a user-defined tolerance and assume that the maximum entry of Equation (27) is greater than . In this case, the segment either needs to be divided into more segments or the degree of

show abstract

hp-adaptive least squares spectral element method for population balance equations

Cited by 27 publications

References 22 publications

Spectral solution of the breakage–coalescence population balance equation Picard and Newton iteration methods

Spectral solution of the breakage–coalescence population balance equation Picard and Newton iteration methods

An Improved Adaptive hp Algorithm using Pseudospectral Methods for Optimal Control

An hp‐adaptive pseudospectral method for solving optimal control problems

Contact Info

Product

Resources

About