| 1. |
- Andersson, Christian, et al.
(författare)
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Derivative-free Parameter Optimization of Functional Mock-up Units
- 2012
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Ingår i: [Host publication title missing].
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Konferensbidrag (refereegranskat)abstract
- Representing a physical system with a mathematical model requires knowledge not only about the physical laws governing the dynamics but also about the parameter values of the system. The parameters can sometimes be measured or calculated, however some of them are often difficult or impossible to obtain in these ways. Finding accurate parameter values is crucial for the accuracy of the mathematical model. Estimating the parameters using optimization algorithms which attempt to minimize the error between the response from the mathematical model and the physical system is a common approach for improving the accuracy of the model. Optimization algorithms usually requires information about the derivatives which may not always be available or not be appropriate for estimation which forces the use of derivative-free optimization algorithms. In this paper, we present an implementation of derivative-free optimization algorithms for parameter estimation in the JModelica.org platform. The implementation allows the underlying dynamic system to be represented as a Functional Mock-up Unit (FMU), thus enables parameter estimation of models designed in modeling tools following the standardized interface, the Functional Mock-up Interface (FMI), such as Dymola.
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| 2. |
- Betancourt, Fernando, et al.
(författare)
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Advanced methods of flux identification for clarifier–thickener simulation models
- 2014
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Ingår i: Minerals Engineering. - : Elsevier BV. - 0892-6875. ; 63:August 2014, s. 2-15
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Tidskriftsartikel (refereegranskat)abstract
- Mathematical models for the simulation of batch settling and continuous clarifier-thickeners can usually be expressed as a convection-diffusion partial differential equation (PDE). Reliable numerical methods require that the nonlinear flux function of this PDE has been identified for a given material. This contribution summarizes, and applies to experimental data, a recent approach [Bürger, R., Diehl, S., 2013. Inverse Problems 29, 045008] for the flux identification in the case of a suspension that shows no compressive behavior. The experimental Kynch test and the Diehl test, which are based on an initially homogenous suspension either filling the whole settling column or being initially located above clear liquid, respectively, provide data points that represent a convex and concave, respectively, suspension-supernate interface. A provably convex (concave) smooth approximation of this interface is obtained by solving a constrained least-squares minimization problem. The interface-approximating function can be converted uniquely into an explicit formula for a convex (concave) part of the flux function.
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| 3. |
- Betancourt, Fernando, et al.
(författare)
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Modeling and controlling clarifier–thickeners fed by suspensions with time-dependent properties
- 2014
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Ingår i: Minerals Engineering. - : Elsevier BV. - 0892-6875. ; 62, s. 91-101
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Tidskriftsartikel (refereegranskat)abstract
- A one-dimensional model of the process of continuous sedimentation in a clarifier–thickener unit is presented. The governing model is expressed as a system of two nonlinear partial differential equations for the solids volume fraction and the varying settling velocity of the solids as functions of depth and time. This model extends the well-known model for the dynamics of a flocculated suspension in a clarifier–thickener advanced by Bürger et al. (2005). Operating charts are calculated to be used for the control of steady states, in particular, to keep the sediment level and the underflow volume fraction at desired values. A numerical scheme and a simple regulator are proposed and numerical simulations are performed.
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| 4. |
- Bürger, Raimund, et al.
(författare)
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A consistent modelling methodology for secondary settling tanks: A reliable numerical method.
- 2013
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Ingår i: Water Science and Technology. - : IWA Publishing. - 0273-1223 .- 1996-9732. ; 68:1, s. 192-208
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Tidskriftsartikel (refereegranskat)abstract
- The consistent modelling methodology for secondary settling tanks (SSTs) leads to a partial differential equation (PDE) of nonlinear convection–diffusion type as a one-dimensional model for the solids concentration as a function of depth and time. This PDE includes a flux that depends discontinuously on spatial position modelling hindered settling and bulk flows, a singular source term describing the feed mechanism, a degenerating term accounting for sediment compressibility, and a dispersion term for turbulence. In addition, the solution itself is discontinuous. A consistent, reliable and robust numerical method that properly handles these difficulties is presented. Many constitutive relations for hindered settling, compression and dispersion can be used within the model, allowing the user to switch on and off effects of interest depending on the modelling goal as well as investigate the suitability of certain constitutive expressions. Simulations show the effect of the dispersion term on effluent suspended solids and total sludge mass in the SST. The focus is on correct implementation whereas calibration and validation are not pursued.
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| 5. |
- Bürger, Raimund, et al.
(författare)
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A consistent modelling methodology for secondary settling tanks in wastewater treatment
- 2011
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Ingår i: Water Research. - : Elsevier BV. - 1879-2448 .- 0043-1354. ; 45:6, s. 2247-2260
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Tidskriftsartikel (refereegranskat)abstract
- The aim of this contribution is partly to build consensus on a consistent modelling methodology (CMM) of complex real processes in wastewater treatment by combining classical concepts with results from applied mathematics, and partly to apply it to the clarification-thickening process in the secondary settling tank. In the CMM, the real process should be approximated by a mathematical model (process model; ordinary or partial differential equation (ODE or PDE)), which in turn is approximated by a simulation model (numerical method) implemented on a computer. These steps have often not been carried out in a correct way. The secondary settling tank was chosen as a case since this is one of the most complex processes in a wastewater treatment plant and simulation models developed decades ago have no guarantee of satisfying fundamental mathematical and physical properties. Nevertheless, such methods are still used in commercial tools to date. This particularly becomes of interest as the state-of-the-art practice is moving towards plant-wide modelling. Then all submodels interact and errors propagate through the model and severely hamper any calibration effort and, hence, the predictive purpose of the model. The CMM is described by applying it first to a simple conversion process in the biological reactor yielding an ODE solver, and then to the solideliquid separation in the secondary settling tank, yielding a PDE solver. Time has come to incorporate established mathematical techniques into environmental engineering, and wastewater treatment modelling in particular, and to use proven reliable and consistent simulation models.
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| 6. |
- Burger, Raimund, et al.
(författare)
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Convexity-preserving flux identification for scalar conservation laws modelling sedimentation
- 2013
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Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 29:4
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Tidskriftsartikel (refereegranskat)abstract
- Sedimentation of a suspension of small particles dispersed in a viscous fluid can be described by a scalar, nonlinear conservation law, whose flux function usually has one inflection point. The identification of the flux function is of theoretical interest and practical importance for plant-scale simulators of continuous sedimentation. For a real suspension, the Kynch test and the Diehl test, which are based on an initially homogenous suspension either filling the whole settling column or being initially located above clear liquid, respectively, provide data points that represent curved (convex or concave, respectively) suspension-supernate interfaces from which it is possible to reconstruct portions of the flux function to either side of the inflection point. Several functional forms can be employed to generate a provably convex or concave, twice differentiable accurate approximation of these data points via the solution of a constrained least-squares minimization problem. The resulting spline-like estimated trajectory can be converted into an explicit formula for the flux function. It is proved that the inverse problem of flux identification solved this way has a unique solution. The problem of gluing together the portions of the flux function from the Kynch and Diehl tests is addressed. Examples involving synthetic data are presented.
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| 7. |
- Bürger, Raimund, et al.
(författare)
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On reliable and unreliable numerical methods for the simulation of secondary settling tanks in wastewater treatment
- 2012
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Ingår i: Computers & Chemical Engineering. - : Elsevier BV. - 1873-4375 .- 0098-1354. ; 41, s. 93-105
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Tidskriftsartikel (refereegranskat)abstract
- Abstract in UndeterminedA one-dimensional model for the sedimentation-compression-dispersion process in the secondary settling tank can be expressed as a nonlinear strongly degenerate parabolic partial differential equation (PDE), which has coefficients with spatial discontinuities. Reliable numerical methods for simulation produce approximate solutions that converge to the physically relevant solution of the PDE as the discretization is refined. We focus on two such methods and assess their performance via simulations for two scenarios. One method is provably convergent and is used as a reference method. The other method is less efficient in reducing numerical errors, but faster and more easily implemented. Furthermore, we demonstrate some pitfalls when deriving numerical methods for this type of PDE and can thereby rule out certain methods as unsuitable; among others, the wide-spread Takacs method. (c) 2012 Elsevier Ltd. All rights reserved.
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| 8. |
- Bürger, Raimund, et al.
(författare)
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On time discretizations for the batch simulation of the settling-compression process in one dimension
- 2016
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Ingår i: Water Science and Technology. - : IWA Publishing. - 0273-1223 .- 1996-9732. ; 73:5, s. 1010-1017
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Tidskriftsartikel (refereegranskat)abstract
- The main purpose of the recently introduced Bürger-Diehl simulation model for secondary settling tanks was to resolve spatial discretization problems when both hindered settling and the phenomena of compression and dispersion are included. Straightforward time integration unfortunately means long computational times. The next step in the development is to introduce and investigate time-integration methods for more efficient simulations, but where other aspects such as implementation complexity and robustness are equally considered. This is done for batch settling simulations. The key findings are partly a new time-discretization method and partly its comparison with other specially tailored and standard methods. Several advantages and disadvantages for each method are given. One conclusion is that the new linearly implicit method is easier to implement than another one (semi-implicit method), but less efficient based on two types of batch sedimentation tests.
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