an improved finite element meshing strategy for dynamic...

Research ArticleAn Improved Finite Element Meshing Strategy forDynamic Optimization Problems

Minliang Gong Aipeng Jiang Quannan Zhang HaokunWang Junjie Hu and Yinghui Lin

School of Automation Hangzhou Dianzi University Hangzhou 310018 China

Correspondence should be addressed to Aipeng Jiang jiangaipeng163com

Received 7 April 2017 Accepted 11 June 2017 Published 19 July 2017

Academic Editor Rahmat Ellahi

Copyright copy 2017 Minliang Gong et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The finite element orthogonal collocation method is widely used in the discretization of differential algebraic equations (DAEs)while the discrete strategy significantly affects the accuracy and efficiency of the results In this work a finite element meshingmethod with error estimation on noncollocation point is proposed and several cases were studied Firstly the simultaneous strategybased on the finite element is used to transform the differential and algebraic optimization problems (DAOPs) into large scalenonlinear programming problems Then the state variables of the reaction process are obtained by simulating with fixed controlvariables The noncollocation points are introduced to compute the error estimates of the state variables at noncollocation pointsFinally in order to improve the computational accuracy with less finite element moving finite element strategy was used fordynamically adjusting the length of finite element appropriately to satisfy the set margin of error The proposed strategy is appliedto two classical control problems and a large scale reverse osmosis seawater desalination process Computing result shows that theproposed strategy can effectively reduce the computing effort with satisfied accuracy for dynamic optimization problems

1 Introduction

The direct transcription method is an important method tosolve the problem of optimal control By discretization ofdifferential and algebraic optimization problems (DAOPs)the state variables and control variables are completelydiscretized The discrete method uses the finite elementorthogonal collocation and generally the number of finiteelements is empirically selected and the length of each finiteelement is equally divided This results in low discretizationaccuracy for state and control variables and to guarantee thesatisfactory accuracy for some problems the calculation timeis too long to accept Moving finite element strategy is a goodidea for the solution With the need of discrete differentialalgebraic equations moving finite element is becoming thepopular and practical technique for chemical process

At present we are concerned with calculation accuracynot only of the material but also of the time in the processparameters for the chemical process attention Modelingmethods based on first principle and data-driven are used forpractical control With the development of solving technol-ogy we can better understand the changes of state variablesin the whole process of chemical reaction

Betts and Kolmanovsky proposed a refinement pro-cedure for nonlinear programming for discrete processesand estimating the discretization error for state variables[1] Liu et al used a novel penalty method to deal withnonlinear dynamic optimization problems with inequalitypath constraints [2] Paiva and Fontes studied the adaptivemesh refinement algorithms which allow a nonuniform nodecollocation and apply a time mesh refinement strategy basedon the local error into practical problems [3] Zhao andTsiotras introduced an efficient and simple method basedon density (or monitor) functions which have been usedextensively for grid refinement The accuracy and stabilityof the method were improved by choosing the appropriatedensity function [4] When using the discrete method tosolve the nonlinear problem iterative programming (IDP)algorithm is rather vulnerable to the stage of time in severalaspects such as accuracy and the convergence rate Li et alintroduce a self-adaptive variable-step IDP algorithm takingaccount of the performance and control variables [5] Zhanget al presented an adaptive variable-step-size RKF methodbased on norm control The method automatically adjuststhe step size and the case where the local error norm value

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 4829195 10 pageshttpsdoiorg10115520174829195

2 Mathematical Problems in Engineering

is 0 or the minimum value is discussed [6] In view ofthe optimization of biochemical processes some researchershave proposed adaptive parameterization methods to solvesuch problems [7 8]

The above work is quite helpful to quick and stablesolution of the dynamic optimization problem but most ofthem put emphasis on method of choice and do not proposea specific operational process As we know the results of theoptimization problem are often dependent on the specificalgorithm

This paper is a mesh refinement strategy based on thevariable finite element mesh method [9ndash11] Prior to opti-mization a suitable initial mesh and length are obtained byGAMS platform simulation Then to optimize the objectivefunction the finite element is moved appropriately under thecondition that the set error is satisfiedThemethod is appliedto the chemical reaction of simple ODE equation [12 13]and large scale reverse osmosis seawater desalination processwhich proves its validity and feasibility

2 Finite Element Discrete Method forEquation Model

According to the whole chemical reaction process we wantto know the changes in the material Discrete solution ofthe model is the key part for computing process Here weintroduce the orthogonal polynomials based on Lagrangersquosuse of the Radau collocation points on the finite element [14]and use of simultaneous method to the entire time domain[0 119905119891] The simplified model can be written as the followingequations

min119911119894119896119906119894119896

120601 (119911 (119905119891))

st 119911119894119896 = 119911119894 + ℎ119894119873119888sum119895=1

Ω(120591119896) 119891 (119911119894119895 119910119894119895 119901)0 = 119892 (119911119894119896 119910119894119896 119906119894119896 119901)119911119894+1 = 1199111198941198731198881199110 = 1199110119911119871 le 119911119894119896 le 119911119880119910119871 le 119910119894119896 le 119910119880119875119871 le 119875 le 119875119880119906119871 le 119906119894119896 le 119906119880119894 = 0 119873 minus 1 119896 = 1 119873119888

(1)

where120601 is the scalar objective function 119911(119905) isin 119877119873119911 and119910(119905) isin119877119873119910 as differential and algebraic variables respectively 119892is the constraint equation for the state variables and thecontrol variables 119906(119905) isin 119877119873119906 and 119901 isin 119877119873119901 as a controlvariable and time independent optimization variable Ω119895(120591119896)is a polynomial of order119873119888+1 in collocation 120591119896 ℎ119894 denotes thelength of element 119894 119911119894119896 is value of its first derivative in element119894 at the collocation point 119896 The advantage of the Lagrangeinterpolation polynomial over other interpolation methods

is that the value of the variable at each collocation point isexactly equal to its coefficient

119905119894119896 = 119905119894minus1 + (119905119894 minus 119905119894minus1) 120591119896119911119896 (119905119894119896) = 119911119894119896119906119896 (119905119894119896) = 119906119894119896

(2)

3 Error Calculation at Noncollocation Point

On the selection of collocation points on finite elements if themathematical expression is configured according to theGausspoint the algebraic precision of the numerical integration isthe highest 119896Gauss points can achieve the algebraic precisionof 2119896 minus 1 order When using 119896 Radau points to configurethe algebraic precision is lower than the Gauss point 2119896 minus2 The discretization proposition based on Radau point hasbetter stability [15] Therefore this paper uses Radau pointsto configure

In order to improve the accuracy of the solution Vasan-tharajan and Biegler [16] inserted the noncollocation pointson the finite element to determine the increase and decreaseof the finite element by the error of the noncollocation point

Generally in the solution of problem (1) it will causecompute error on state or control profiles due to finite meshIn this study the error of collocation finite elements can beestimated from

119879119894 (119905nc) = 119889119911 (119905nc)119889120591 minus ℎ119894119891nc (3)

error (119894) = 119862119879119894 (119905nc) (4)

Here 119862 is constant depending on collocation points 119891nc isdifferential state equation at 119905nc

119862 = 1119860 int120591nc

0

119870prod119895=1

(119904 minus 120591119895) 119889119904

119860 = 119870prod119895=1

(120591nc minus 120591119895) (5)

From (3) 119879119894 will be zero at the collocations So we needto select a noncollocation point 120591nc that is 119905nc = 119905119894 + ℎ119894120591ncSpecifically 119889119911(119905nc)119889120591 can be stated as

119889119911 (119905nc)119889120591 = ℎ119894119873119862sum119895=1

Ω (120591nc) 119891 (6)

where Ω(120591nc) is the differential ofΩ119895(120591119896)Thenwe add (3)-(4)to problem (1) and then resolve the OPC problem

4 Finite Element Method

As the error estimation gotten with (3) and (4) we can dividethe finite elements mesh Generally equally spaced methodwaswidely used inmany research due to its simply operationThe flow chart of calculation is shown in Figure 1

Mathematical Problems in Engineering 3

Solving optimal controlproblem with IPOPT

The estimation of thefinite element N iscompleted

Yes

No

Discretized optimalcontrol problem

Set the number of initialfinite elements N andlength ℎi

Rese

ts th

e fini

teel

emen

t N an

d th

ele

ngth

ℎi

If min(error) lt 120576

Figure 1 Equally spaced method for finite element mesh

On the contrary this approach will take more time andget larger 119873 after the mesh divided Now we take an idea ofadaptive mesh and obtain algorithm of mesh initialization

Step 1 Divide the time domain [1199050 119905119891] equally with a coarsemesh and compute the error at every location of element bysolving problem (1)

Step 2 When each finite element error radio119894 = log(|error119894120576|) lt 05 end the initialization of the finite element If therequirement of less than 05 is not met turn to Step 3

Step 3 Once the error of each finite element radio119894 ge 05 add119877 finite element here 119877 rounds up from radio119894 Then resolveproblem (1)

However the finite element mesh gotten above has notsatisfied accuracy yet So we first fix the mesh ℎ119894 and suitablecontrol profile and take simulate calculation by solvingproblem (1) without objective function After that relax ℎ119894and add constraint (7) as follows

119873sum119894minus1

ℎ119894 = 119905119891 ℎ119894 isin [06ℎ119894 14ℎ119894] (7)

Give the initial estimate ofthe finite element N

Solve the optimization problemand get local error

No

Yes

Discretized optimal controlproblem

Rese

t the

fini

teel

emen

t with

the

part

ition

crite

rion

If all radiＩ lt 05

radioi = ＦＩＡ(儨儨儨儨errori)

modelRelease hi and resolve the

Output the length of each finite element hi Then the initial values of each variable are obtained by fixing the length of each finite element These initial values are used to optimize the target values of the problem

Figure 2 Finite element mesh refinement algorithm chart

Through this way seasonable initial values for optimiza-tion are provided What is more it also offers some freedomto search the breakpoint of control profiles

From now on we can summarize the approach describedwith program flow chart in Figure 2

Through the above grid division and GAMS platformsimulation and optimization the optimized result is obtainedfinally However the optimality of the above result should beverified According to the optimal control theory the value ofthe Hamiltonian function comprising the objective functionand the constraints should be kept as constant along the timeaxis According to the optimal control theory the discreteHamiltonian function is denoted as

119867119894119895 = 120582119894119895119879119891 (119911119894119895 119910119894119895 119906119894119895 119901)+ 120583119894119895119879119892119864 (119911119894119895 119910119894119895 119906119894119895 119901)+ 120574119894119895119879119892119894 (119911119894119895 119910119894119895 119906119894119895 119901)

(8)


When the solution is optimal the last two terms ofthe Hamiltonian are zero According to the optimal controltheory when a system is optimal the Hamiltonian must be aconstant

119867119894119895 = Constant forall119894 119895 (9)

Therefore if the finite element and the configurationpoint are not constant the result is not optimal and themeshing needs to readd the finite element Here we usemethod of Tanartkit and Biegler [11] who put forward aHamilton function criterion

10038171003817100381710038171003817100381710038171003817100381710038171003817119889119867 (119905119894119895)

11988911990510038171003817100381710038171003817100381710038171003817100381710038171003817le tol (10)

When tol is set to allowable errors it is set asmax119894119895(00010001119867119894119895) If the meshing method proposed by us can notsatisfy the above Hamilton function error requirement wemust readd the finite element mesh quantity

5 Case Study

This numerical experiment is based on Intel (R) Core (TM)i3-2350M 23GHz processor with 4GB RAM the nonlinearprogramming solver IPOPT [17 18] is developed by CarnegieMellon University

To illustrate the above strategies we consider two classicalprocess dynamic optimization problems and a large scalereverse osmosis seawater (SWRO) desalination process Atthe same time we use the error analysis based on theequipartition finite element method for comparison For thefollowing example we use the 3-order Radau collocationpoints each finite element has a noncollocation point Threeof these collocation points are 1205911 = 0155051 1205912 = 0644949and 1205913 = 1 noncollocation point is 120591nc = 05 finite elementtolerance error is 120576 = 10119890 minus 5 and default set point errorestimation limit is 120576 = 10119890 minus 551 Batch Temperature Profile with Parallel Reactions Thisexample is a chemical process considering a nonisothermalreactor with first order 119860 rarr 119861 119860 rarr 119862 where 119860 is reactionmaterial 119861 is target product and 119862 is by-product Now thegoal is to find a feasible temperature profile that maximizesthe final amount of product 119861 after one hour The optimalcontrol problem can be stated as

min minus 119887 (1)st 119889119886

119889119905 = minus119886 (119905) (119906 (119905) + 119906 (119905)22 )

119889119887119889119905 = 119886 (119905) 119906 (119905)119886 (0) = 1119887 (0) = 00 le 119905 le 1 119906 (119905) isin [0 5]

(11)

Permissible errorab

2 3 4 5 6 7 8 9 10 11 12 13 141Location of finite element

0

05

1

15

Erro

r

times10minus5

Figure 3 Error of differential profiles at finite elements with equallyspaced method

2 3 4 5 6 71Location of finite element

0

05

1

15Er

ror

times10minus5

Permissible errorab

Figure 4 Error of differential profiles at finite elements with newmethod

The problem has been solved in many pieces of literatureIn [19] adaptive iterative dynamic programming leads to 50time steps and needs more than 60CPUs Here we first adoptequally spaced elements method which leads to 14 finiteelements Local errors for all finite elements are shown inFigure 3

As shown in Figure 3 we can see that the first- andsecond-finite element errors are large and other places haveto meet the set error tolerance This means that we can usefewer finite elements to calculate the problem In Figure 4when using our proposed meshing method only 7 finiteelements are needed In the use of the equally finite elementmethod the posterior finite element error is very small sothat when we use a partitioning method the error is close tothe set error tolerance

In Figure 5 fixed finite element length is gotten from thesimulation After adding the boundary constraint (7) a new


0

01

02

03

04Le

ngth

of fi

nite

elem

ent

2 3 4 5 6 71Location finite element

Fixed ℎiVar hi

Figure 5 Size of finite element in simulation and optimization

0

1

2

3

4

5

Con

trol p

rofil

e (U

)

02 04 06 08 10Time (hour)

Figure 6 Optimal control for the batch reaction problem

ab

0

02

04

06

08

1

Stat

e pro

file

02 04 06 08 10Time (hour)

Figure 7 State profiles for the batch reaction problem

finite element length is obtained (blue line)This can also helpfor locating breakpoints into specific problems

Figures 6 and 7 show the optimal control image and thestate variable change image respectively These changes areclose to those obtained by previous researchers This fullyproves the feasibility of the proposed method

005

01

015

02

025

Ham

ilton

ian

02 04 06 08 10Time (hour)

Figure 8 Profile of Hamiltonian function

Table 1 Batch reaction numerical results

Strategy 119873 Var Eq Iter Accuracy CPU(s)ESM 14 294 252 11 1119890 minus 5 9841MFE 7 217 190 39 1119890 minus 5 5330

Figure 8 shows that the Hamilton function is a constantand the derivative of Hamiltonian function satisfies the aboveerror that the problem has been the optimal solution

From Table 1 we can see that by using an improved finiteelement method we propose the finite element number canbe reduced by half The calculation time can be reduced by45 s under the same precision

52 Rayleigh Problem Here we consider another optimalcontrol problem which can be described as

min int119905119891=250

(11990912 + 1199062) 119889119905st 1198891199091119889119905 = 1199092

1198891199092119889119905 = minus1199091 + (14 minus 01411990922) 1199092 + 41199061199091 (0) = minus51199092 (0) = minus5

(12)

In this example the number of finite elements is oftenestimated by trial and error Here we use 104 equally spacedfinite elements to make the two differential equations meetthe error requirements as shown in Figure 9

As can be seen from Figure 9 most of the finite elementshave been early to meet the error requirements Here weuse the new method we propose Only 8 finite elementsare required to satisfy the error requirements as shown inFigure 10

Figure 11 shows the length of each finite element positionIt can be seen from the figure that the length of the 6 7and 8th finite element position is greater than the length ofthe simulation (red line) which indicates that the addition


Permissible errorx1

x2

11 21 31 41 51 61 71 81 91 1011Location of finite element

Erro

r

0

05

1

15times10

minus5


Permissible errorx1

x2

2 3 4 5 6 7 81Location of finite element

0

05

1

15

Erro

r

times10minus5


Table 2 Rayleigh reaction numerical results


of the boundary constraint is effective when optimizing theobjective function and finding the boundary breakpoint

Figures 12 and 13 show the optimal control image and thestate variable change image in process

The Hamiltonian function image of Figure 14 indicatesthat the objective function has been optimized

InTable 2 whenusing the finite elementmethod the totalnumber of variables is 36 and the computer time is 67277 sBut using our new method the total number of discretevariables is 8 and the calculation time is 5352 s In terms of

Fixed ℎiVar ℎi

0

01

02

03

04

Leng

th o

f fini

te el

emen

t



0

1

2

3

4

5

Con

trol p

rofil

e (U

)

05 1 15 2 250Time

Figure 12 Optimal control for Rayleigh problem

x1

x2

05 1 15 2 250Time

minus10

minus5

0

5

10

Stat

e pro

file

Figure 13 State profiles for Rayleigh problem

the number of finite elements and the computation time thenew method presented in this paper has some advantages

53 Reverse Osmosis Sweater Desalination Process The fea-sibility of the proposed method is illustrated by two classicalchemical reaction control problemsThe following is an oper-ational optimization study of a large scale reverse osmosis


minus60

minus50

minus40

minus30

minus20

minus10

0

10

Ham

ilton

ian

05 1 15 2 250Time


seawater (SWRO) desalination process The model equationcan be expressed as

119860119908 (119894 119895) = 1198601199080 sdot 119865119860 sdot 119872119865119860sdot 119890(1205721((119879minus273)273)minus1205722(119875119891minus119875119889(119894119895)))

119861119904 = 1198611199040 sdot 119865119861 sdot 119872119865119861 sdot 119890(1205731((119879minus273)273))119869V119894119895 = 119860119908 (119875119891 minus 119875119889119894119895 minus Δ120587119894119895) 119869119904119894119895 = 119861119908 (119862119898119894119895 minus 119862119901119894119895) 119862119901119894119895 = 119869119904119894119895

119869V119894119895 Δ120587119894119895 = RT (119862119898119894119895 minus 119862119901119894119895) 119862119898119894119895 = [(119862119887119894119895 minus 119862119901119894119895) exp(119869V119894119895

119896119888119894119895 )] + 119862119901119894119895

119896119888119894119895 = 0065Re1198941198950875Sc119894119895025 sdot 119889119890119863119860119861 Sc119894119895 = 120583119894119895

(120588119863119860119861) 120582119894119895 = 623119870120582Reminus03119894119895 Re119894119895 = 120588119881119894119895119889119890

120583119894119895 119889119881119894119895119889119911 = minus2119869V119894119895ℎsp

119889119862119887119894119895119889119911 = 2119869V119894119895

ℎsp119881119894119895 (119862119887119894119895 minus 119862119901119894119895) 119889119875119889119894119895119889119911 = minus120582119894119895 120588119889119890

11988121198941198952

(13)

And then according to the objective function of the cost ofwater production

min119876119891119875119891119867119905119879

OC = OCIP +OCEN +OCCH +OCOTR (14)

Here OCIP and OCEN are reverse osmosis process waterintake and RO energy costs OCCH is the cost of systemchemicals OCOTR indicates other costs including labor costsand maintenance costs

The optimization proposition also needs to satisfy theRO process model the reservoir model the operation costmodel and the inequality constraint equation The systemcost model is shown as follows

OCEN = SEC sdot 119876119901 sdot 119875elcOCIP = 119875in sdot 119876119891 sdot 119875elc sdot 119875LF120578IP OCCH = 00225119876119891

(15)

Here 119875elc is the unit price 120578IP is the motor energyefficiency 119875LF represents the load factor 119875in is the outletpressure of the pump

From the optimization program mainly to the day thereare peaks and valleys of electricity prices resulting in differentperiods of water production costs The system can be donethrough the cistern reservoir volume regulating the watersystem and the userrsquos demand for water to reduce the overallcost

The reverse osmosis process model mainly includes threedifferential variablesmembrane concentration flow rate andpressure In this paper the number of finite elements isdivided into discrete errors based on the steady-state simu-lation process so as to reduce the computational complexityof the optimization model

Figure 15 shows three differential variables and the errorof velocity concentration and pressure can only meet theerror tolerance when the 22 finite elements are dividedequally Since the error of velocity and pressure is muchsmaller than that of channel concentration the discrete erroranalysis was mainly focused on the channel concentrationFigure 16 is the finite element error analysis using the newmethod

Figure 17 shows that after the initial finite element lengthis obtained by simulation the constraint equation is added toobtain a more suitable finite element length

Figures 18 19 and 20 represent the flow rate concen-tration and pressure at the feed of the membrane modulerespectively It can be seen from these figures that addingboundary constraints to each curve is more smoothly andmore responsive to changes in the variables within the actualmembrane module

Table 3 shows the use of a method proposed in this paperboth from the number of variables the number of iterationsand computing time above a great upgrade

It can be seen that the number of finite elements requiredfor the model discretization is reduced to 14 when usingthe moving finite element method Due to the finite elementmethod the finite element length can be adjusted to a smallextent which greatly increases the accuracy of the solution


VelocityConcentrationPressure

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 221Location of finite element

0

02

04

06

08

1

12

Erro

r

times10minus5

Figure 15 Error of differential profiles at finite elementswith equallyspacedmethodThe black dotted line represents themaximum errorof each finite element at the noncollocation point

times10minus5

0

02

04

06

08

1

12

Erro

r



Figure 16 Error of differential profiles at finite elements with newmethodThe black dotted line represents themaximumerror of eachfinite element at the noncollocation point

6 Conclusion

In this paper we proposed amesh-partitioning strategy basedon the direct transcription method to solve the optimalcontrol problem This method discretizes the differentialalgebraic equation (DAE) using the Radau collocation pointbased on the variable finite element and finally transforms

Fixed hiVar hi

0

02

04

06

08

1

Leng

th o

f fini

te el

emen

t



Fixed ℎiVar ℎi

016

018

02

022

024Ve

loci

ty (m

s)


Figure 18 Membrane channel within one hour of the velocitychanges

Table 3 SWRO numerical results


into a nonlinear programming problem Here the initializa-tion variable finite element uses a valid termination criterionAt the same time the computation time can be saved underthe condition of satisfying certain precision

Cases study of two classical control problems and alarge scale reverse osmosis process optimization problemwas carried with our proposed method and conventionalmethod The computing results show that the proposedmethod can effectively reduce the number of finite elementand thus reduce the computing effortswith permitted discreteaccuracy


Fixed hiVar hi


30

35

40

45C

once

ntra

tion

(kg

m3)

Figure 19Membrane channel within one hour of the concentrationchanges

Fixed hiVar hi

0

01

02

03

04

05

Pres

sure

(bar

)


Figure 20 Membrane channel within one hour of the pressurechanges

Nomenclature

119905119891 Terminal time (s)120601 The scalar objective function119911(119905) Differential state variables119910(119905) Algebraic state variables119892 Constraint equation for the state and controlvariables119906(119905) Control variable119901 Time independent optimization variableℎ119894 The length of element 119894119896 Number of collocation points119911119894119896 First derivative in element 119894 at the collocationpoint 119896119888 Constant depending on collocation points119891nc Differential state equation at 119905nc120576 Default set point error estimation limit

tol The setting error119873 Number of finite elementsVar Number of variablesEq Number of equations

Iter The number of iterations calculatedESM Equally spaced methodMFE Moving finite elements119860119908 Membrane water permeability (msdotsminus1sdotPaminus1)1198601199080 Intrinsic membrane water permeability

(msdotsminus1sdotPaminus1)119861119904 Membrane TDS permeability (ms)1198611199040 Intrinsic membrane TDS permeability (ms)1205721 Constant parameter1205722 Constant parameter1205731 Constant parameter119879 Feed (operational) temperature (K)119875119891 Feed pressure (bar)119875119889 Pressure drop along RO spiral wound module(bar)119869V Solvent flux (kgm2sdots)Δ120587 Pressure loss of osmosis pressure (bar)119869119904 Solute flux (kgm2sdots)119862119898 Salt concentration of membrane surface (kgm3)119862119901 Permeate concentration of RO unit (kgm3)119862119887 Bulk concentration along feed channel (kgm3)119896119888 Mass transfer coefficient (ms)

Re Reynolds number (dimensionless)Sc Schmidt number (dimensionless)119889119890 Hydraulic diameter of the feed spacer channel (m)119863119860119861 Dynamic viscosity (m2s)120583 Kinematic viscosity120588 Density of permeate water (kgm3)120582 Friction factor119870120582 Empirical parameters119881 Axial velocity in feed channel (ms)ℎsp Height of the feed spacer channel (m)OC Operational cost of entire processOCEN Energy cost of RO processOCIP Energy cost for intake and pretreatmentOCCH Cost for chemicalsOCOTR Other costsSEC Specific energy consumption (kwsdothm3)119876119901 Permeate flow rate (m3h)119875elc Electricity price (CNYkwsdoth)119875in Output pressure of the intake pump (bar)PLF Load coefficient120578IP Mechanical efficiency of intake pump

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

This work is supported by National Natural Science Foun-dation (NNSF) of China (no 61374142) the Natural ScienceFoundation of Zhejiang (LY16F030006) the Public Projectsof Zhejiang Province China (no 2017C31065) Researchand Innovation Fund of Hangzhou Dianzi University(CXJJ2016043) and Hangzhou Dianzi University GraduateCore Curriculum Construction Project (GK168800299024-012)


References

[1] J T Betts and I Kolmanovsky Practical methods for optimalcontrol using nonlinear programming [MS thesis] Society forIndustrial and Applied Mathematics (SIAM) Philadelphia PAUSA 2001

[2] X Liu Y Hu J Feng and K Liu ldquoA novel penalty approach fornonlinear dynamic optimization problems with inequality pathconstraintsrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 59 no 10 pp 2863ndash2867 2014

[3] L T Paiva and F A C C Fontes ldquoTimemdashmesh Refinementin Optimal Control Problems for Nonholonomic VehiclesrdquoProcedia Technology vol 17 pp 178ndash185 2014

[4] Y Zhao and P Tsiotras ldquoDensity functions for mesh refinementin numerical optimal controlrdquo Journal of Guidance Control andDynamics vol 34 no 1 pp 271ndash277 2011

[5] H-G Li J-P Liu and J-W Huang ldquoSelf-adaptive variable-stepapproach for iterative dynamic programming with applicationsin batch process optimizationrdquoControl andDecision vol 30 no11 pp 2048ndash2054 2015

[6] Z Zhang Y Liao and Y Wen ldquoAn adaptive variable step-sizeRKF method and its application in satellite orbit predictionrdquoComputer Engineering and Science vol 37 no 4 pp 842ndash8462015

[7] P Liu G Li X Liu and Z Zhang ldquoNovel non-uniformadaptive grid refinement control parameterization approach forbiochemical processes optimizationrdquo Biochemical EngineeringJournal vol 111 pp 63ndash74 2016

[8] L Wang X Liu and Z Zhang ldquoA new sensitivity-basedadaptive control vector parameterization approach for dynamicoptimization of bioprocessesrdquo Bioprocess and Biosystems Engi-neering vol 40 no 2 pp 181ndash189 2017

[9] L T Biegler Nonlinear programming concepts algorithmsand applications to chemical processes [MS thesis] Society forIndustrial and AppliedMathematics Pittsburgh Pa USA 2010

[10] W F Chen K X Wang Z J Shao and L T BieglerldquoMoving finite elements for dynamic optimization with directtranscription formulationsrdquo in Control and Optimization withDifferential-Algebraic Constraints L T Biegler S L Campbelland V Mehrmann Eds SIAM Philadelphia PA USA 2011

[11] P Tanartkit and L T Biegler ldquoA nested simultaneous approachfor dynamic optimization problemsmdashII the outer problemrdquoComputers and Chemical Engineering vol 21 no 12 pp 735ndash741 1997

[12] G A Hicks and W H Ray ldquoApproximation methods foroptimal control synthesisrdquo The Canadian Journal of ChemicalEngineering vol 49 no 4 pp 522ndash528 1971

[13] J S Logsdon and L T Biegler ldquoAccurate solution of differential-algebraic optimization problemsrdquo Industrial and EngineeringChemistry Research vol 28 no 11 pp 1628ndash1639 1989

[14] V M Zavala Computational strategies for the optimal operationof large-scale chemical processes [PhD thesis] Carnegie MellonUniversity Pennsylvania Pa USA 2008

[15] J Bausa ldquoDynamic optimization of startup and load-increasingprocesses in power plantsmdashpart I methodrdquo Journal of Engineer-ing for Gas Turbines and Power vol 123 no 1 pp 251ndash254 2001

[16] S Vasantharajan and L T Biegler ldquoSimultaneous strategies foroptimization of differential-algebraic systemswith enforcementof error criteriardquo Computers and Chemical Engineering vol 14no 10 pp 1083ndash1100 1990

[17] A Wachter and L T Biegler ldquoLine search filter methods fornonlinear programming motivation and global convergencerdquoSIAM Journal on Optimization vol 16 no 1 pp 1ndash31 2005

[18] A Chter and L T Biegler ldquoLine search filter methods fornonlinear programming local convergencerdquo SIAM Journal onOptimization vol 16 no 1 pp 32ndash48 2005

[19] L I Qianxing X Liu andW U Gaohui ldquoAn adaptive step-sizeapproach to iterative dynamic programmingrdquoBulletin of Scienceand Technology vol 26 no 5 pp 666ndash669 2010

Submit your manuscripts athttpswwwhindawicom

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MathematicsJournal of


Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of


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International Journal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


is 0 or the minimum value is discussed [6] In view ofthe optimization of biochemical processes some researchershave proposed adaptive parameterization methods to solvesuch problems [7 8]

The above work is quite helpful to quick and stablesolution of the dynamic optimization problem but most ofthem put emphasis on method of choice and do not proposea specific operational process As we know the results of theoptimization problem are often dependent on the specificalgorithm

This paper is a mesh refinement strategy based on thevariable finite element mesh method [9ndash11] Prior to opti-mization a suitable initial mesh and length are obtained byGAMS platform simulation Then to optimize the objectivefunction the finite element is moved appropriately under thecondition that the set error is satisfiedThemethod is appliedto the chemical reaction of simple ODE equation [12 13]and large scale reverse osmosis seawater desalination processwhich proves its validity and feasibility

2 Finite Element Discrete Method forEquation Model

According to the whole chemical reaction process we wantto know the changes in the material Discrete solution ofthe model is the key part for computing process Here weintroduce the orthogonal polynomials based on Lagrangersquosuse of the Radau collocation points on the finite element [14]and use of simultaneous method to the entire time domain[0 119905119891] The simplified model can be written as the followingequations

min119911119894119896119906119894119896

120601 (119911 (119905119891))

st 119911119894119896 = 119911119894 + ℎ119894119873119888sum119895=1

Ω(120591119896) 119891 (119911119894119895 119910119894119895 119901)0 = 119892 (119911119894119896 119910119894119896 119906119894119896 119901)119911119894+1 = 1199111198941198731198881199110 = 1199110119911119871 le 119911119894119896 le 119911119880119910119871 le 119910119894119896 le 119910119880119875119871 le 119875 le 119875119880119906119871 le 119906119894119896 le 119906119880119894 = 0 119873 minus 1 119896 = 1 119873119888

(1)

where120601 is the scalar objective function 119911(119905) isin 119877119873119911 and119910(119905) isin119877119873119910 as differential and algebraic variables respectively 119892is the constraint equation for the state variables and thecontrol variables 119906(119905) isin 119877119873119906 and 119901 isin 119877119873119901 as a controlvariable and time independent optimization variable Ω119895(120591119896)is a polynomial of order119873119888+1 in collocation 120591119896 ℎ119894 denotes thelength of element 119894 119911119894119896 is value of its first derivative in element119894 at the collocation point 119896 The advantage of the Lagrangeinterpolation polynomial over other interpolation methods

is that the value of the variable at each collocation point isexactly equal to its coefficient

119905119894119896 = 119905119894minus1 + (119905119894 minus 119905119894minus1) 120591119896119911119896 (119905119894119896) = 119911119894119896119906119896 (119905119894119896) = 119906119894119896

(2)

3 Error Calculation at Noncollocation Point

On the selection of collocation points on finite elements if themathematical expression is configured according to theGausspoint the algebraic precision of the numerical integration isthe highest 119896Gauss points can achieve the algebraic precisionof 2119896 minus 1 order When using 119896 Radau points to configurethe algebraic precision is lower than the Gauss point 2119896 minus2 The discretization proposition based on Radau point hasbetter stability [15] Therefore this paper uses Radau pointsto configure

In order to improve the accuracy of the solution Vasan-tharajan and Biegler [16] inserted the noncollocation pointson the finite element to determine the increase and decreaseof the finite element by the error of the noncollocation point

Generally in the solution of problem (1) it will causecompute error on state or control profiles due to finite meshIn this study the error of collocation finite elements can beestimated from

119879119894 (119905nc) = 119889119911 (119905nc)119889120591 minus ℎ119894119891nc (3)

error (119894) = 119862119879119894 (119905nc) (4)

Here 119862 is constant depending on collocation points 119891nc isdifferential state equation at 119905nc

119862 = 1119860 int120591nc

0

119870prod119895=1

(119904 minus 120591119895) 119889119904

119860 = 119870prod119895=1

(120591nc minus 120591119895) (5)

From (3) 119879119894 will be zero at the collocations So we needto select a noncollocation point 120591nc that is 119905nc = 119905119894 + ℎ119894120591ncSpecifically 119889119911(119905nc)119889120591 can be stated as

119889119911 (119905nc)119889120591 = ℎ119894119873119862sum119895=1

Ω (120591nc) 119891 (6)

where Ω(120591nc) is the differential ofΩ119895(120591119896)Thenwe add (3)-(4)to problem (1) and then resolve the OPC problem

4 Finite Element Method

As the error estimation gotten with (3) and (4) we can dividethe finite elements mesh Generally equally spaced methodwaswidely used inmany research due to its simply operationThe flow chart of calculation is shown in Figure 1




Yes

No



Rese

ts th

e fini

teel

emen

t N an

d th

ele

ngth

ℎi








119873sum119894minus1

ℎ119894 = 119905119891 ℎ119894 isin [06ℎ119894 14ℎ119894] (7)



No

Yes


Rese

t the

fini

teel

emen

t with

the

part

ition

crite

rion









119867119894119895 = 120582119894119895119879119891 (119911119894119895 119910119894119895 119906119894119895 119901)+ 120583119894119895119879119892119864 (119911119894119895 119910119894119895 119906119894119895 119901)+ 120574119894119895119879119892119894 (119911119894119895 119910119894119895 119906119894119895 119901)

(8)



119867119894119895 = Constant forall119894 119895 (9)


10038171003817100381710038171003817100381710038171003817100381710038171003817119889119867 (119905119894119895)

11988911990510038171003817100381710038171003817100381710038171003817100381710038171003817le tol (10)


5 Case Study



min minus 119887 (1)st 119889119886

119889119905 = minus119886 (119905) (119906 (119905) + 119906 (119905)22 )

119889119887119889119905 = 119886 (119905) 119906 (119905)119886 (0) = 1119887 (0) = 00 le 119905 le 1 119906 (119905) isin [0 5]

(11)

Permissible errorab


0

05

1

15

Erro

r

times10minus5



0

05

1

15Er

ror

times10minus5

Permissible errorab






0

01

02

03

04Le

ngth

of fi

nite

elem

ent


Fixed ℎiVar hi


0

1

2

3

4

5

Con

trol p

rofil

e (U

)

02 04 06 08 10Time (hour)


ab

0

02

04

06

08

1

Stat

e pro

file

02 04 06 08 10Time (hour)




005

01

015

02

025

Ham

ilton

ian

02 04 06 08 10Time (hour)







min int119905119891=250

(11990912 + 1199062) 119889119905st 1198891199091119889119905 = 1199092


(12)





Permissible errorx1

x2


Erro

r

0

05

1

15times10

minus5


Permissible errorx1

x2


0

05

1

15

Erro

r

times10minus5








Fixed ℎiVar ℎi

0

01

02

03

04

Leng

th o

f fini

te el

emen

t



0

1

2

3

4

5

Con

trol p

rofil

e (U

)

05 1 15 2 250Time


x1

x2

05 1 15 2 250Time

minus10

minus5

0

5

10

Stat

e pro

file





minus60

minus50

minus40

minus30

minus20

minus10

0

10

Ham

ilton

ian

05 1 15 2 250Time






119896119888119894119895 )] + 119862119901119894119895

119896119888119894119895 = 0065Re1198941198950875Sc119894119895025 sdot 119889119890119863119860119861 Sc119894119895 = 120583119894119895

(120588119863119860119861) 120582119894119895 = 623119870120582Reminus03119894119895 Re119894119895 = 120588119881119894119895119889119890

120583119894119895 119889119881119894119895119889119911 = minus2119869V119894119895ℎsp

119889119862119887119894119895119889119911 = 2119869V119894119895

ℎsp119881119894119895 (119862119887119894119895 minus 119862119901119894119895) 119889119875119889119894119895119889119911 = minus120582119894119895 120588119889119890

11988121198941198952

(13)


min119876119891119875119891119867119905119879





(15)












0

02

04

06

08

1

12

Erro

r

times10minus5


times10minus5

0

02

04

06

08

1

12

Erro

r




6 Conclusion


Fixed hiVar hi

0

02

04

06

08

1

Leng

th o

f fini

te el

emen

t



Fixed ℎiVar ℎi

016

018

02

022

024Ve

loci

ty (m

s)








Fixed hiVar hi


30

35

40

45C

once

ntra

tion

(kg

m3)


Fixed hiVar hi

0

01

02

03

04

05

Pres

sure

(bar

)



Nomenclature








Acknowledgments



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







Yes

No



Rese

ts th

e fini

teel

emen

t N an

d th

ele

ngth

ℎi








119873sum119894minus1

ℎ119894 = 119905119891 ℎ119894 isin [06ℎ119894 14ℎ119894] (7)



No

Yes


Rese

t the

fini

teel

emen

t with

the

part

ition

crite

rion









119867119894119895 = 120582119894119895119879119891 (119911119894119895 119910119894119895 119906119894119895 119901)+ 120583119894119895119879119892119864 (119911119894119895 119910119894119895 119906119894119895 119901)+ 120574119894119895119879119892119894 (119911119894119895 119910119894119895 119906119894119895 119901)

(8)



119867119894119895 = Constant forall119894 119895 (9)


10038171003817100381710038171003817100381710038171003817100381710038171003817119889119867 (119905119894119895)

11988911990510038171003817100381710038171003817100381710038171003817100381710038171003817le tol (10)


5 Case Study



min minus 119887 (1)st 119889119886

119889119905 = minus119886 (119905) (119906 (119905) + 119906 (119905)22 )

119889119887119889119905 = 119886 (119905) 119906 (119905)119886 (0) = 1119887 (0) = 00 le 119905 le 1 119906 (119905) isin [0 5]

(11)

Permissible errorab


0

05

1

15

Erro

r

times10minus5



0

05

1

15Er

ror

times10minus5

Permissible errorab






0

01

02

03

04Le

ngth

of fi

nite

elem

ent


Fixed ℎiVar hi


0

1

2

3

4

5

Con

trol p

rofil

e (U

)

02 04 06 08 10Time (hour)


ab

0

02

04

06

08

1

Stat

e pro

file

02 04 06 08 10Time (hour)




005

01

015

02

025

Ham

ilton

ian

02 04 06 08 10Time (hour)







min int119905119891=250

(11990912 + 1199062) 119889119905st 1198891199091119889119905 = 1199092


(12)





Permissible errorx1

x2


Erro

r

0

05

1

15times10

minus5


Permissible errorx1

x2


0

05

1

15

Erro

r

times10minus5








Fixed ℎiVar ℎi

0

01

02

03

04

Leng

th o

f fini

te el

emen

t



0

1

2

3

4

5

Con

trol p

rofil

e (U

)

05 1 15 2 250Time


x1

x2

05 1 15 2 250Time

minus10

minus5

0

5

10

Stat

e pro

file





minus60

minus50

minus40

minus30

minus20

minus10

0

10

Ham

ilton

ian

05 1 15 2 250Time






119896119888119894119895 )] + 119862119901119894119895

119896119888119894119895 = 0065Re1198941198950875Sc119894119895025 sdot 119889119890119863119860119861 Sc119894119895 = 120583119894119895

(120588119863119860119861) 120582119894119895 = 623119870120582Reminus03119894119895 Re119894119895 = 120588119881119894119895119889119890

120583119894119895 119889119881119894119895119889119911 = minus2119869V119894119895ℎsp

119889119862119887119894119895119889119911 = 2119869V119894119895

ℎsp119881119894119895 (119862119887119894119895 minus 119862119901119894119895) 119889119875119889119894119895119889119911 = minus120582119894119895 120588119889119890

11988121198941198952

(13)


min119876119891119875119891119867119905119879





(15)












0

02

04

06

08

1

12

Erro

r

times10minus5


times10minus5

0

02

04

06

08

1

12

Erro

r




6 Conclusion


Fixed hiVar hi

0

02

04

06

08

1

Leng

th o

f fini

te el

emen

t



Fixed ℎiVar ℎi

016

018

02

022

024Ve

loci

ty (m

s)








Fixed hiVar hi


30

35

40

45C

once

ntra

tion

(kg

m3)


Fixed hiVar hi

0

01

02

03

04

05

Pres

sure

(bar

)



Nomenclature








Acknowledgments



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119867119894119895 = Constant forall119894 119895 (9)


10038171003817100381710038171003817100381710038171003817100381710038171003817119889119867 (119905119894119895)

11988911990510038171003817100381710038171003817100381710038171003817100381710038171003817le tol (10)


5 Case Study



min minus 119887 (1)st 119889119886

119889119905 = minus119886 (119905) (119906 (119905) + 119906 (119905)22 )

119889119887119889119905 = 119886 (119905) 119906 (119905)119886 (0) = 1119887 (0) = 00 le 119905 le 1 119906 (119905) isin [0 5]

(11)

Permissible errorab


0

05

1

15

Erro

r

times10minus5



0

05

1

15Er

ror

times10minus5

Permissible errorab






0

01

02

03

04Le

ngth

of fi

nite

elem

ent


Fixed ℎiVar hi


0

1

2

3

4

5

Con

trol p

rofil

e (U

)

02 04 06 08 10Time (hour)


ab

0

02

04

06

08

1

Stat

e pro

file

02 04 06 08 10Time (hour)




005

01

015

02

025

Ham

ilton

ian

02 04 06 08 10Time (hour)







min int119905119891=250

(11990912 + 1199062) 119889119905st 1198891199091119889119905 = 1199092


(12)





Permissible errorx1

x2


Erro

r

0

05

1

15times10

minus5


Permissible errorx1

x2


0

05

1

15

Erro

r

times10minus5








Fixed ℎiVar ℎi

0

01

02

03

04

Leng

th o

f fini

te el

emen

t



0

1

2

3

4

5

Con

trol p

rofil

e (U

)

05 1 15 2 250Time


x1

x2

05 1 15 2 250Time

minus10

minus5

0

5

10

Stat

e pro

file





minus60

minus50

minus40

minus30

minus20

minus10

0

10

Ham

ilton

ian

05 1 15 2 250Time






119896119888119894119895 )] + 119862119901119894119895

119896119888119894119895 = 0065Re1198941198950875Sc119894119895025 sdot 119889119890119863119860119861 Sc119894119895 = 120583119894119895

(120588119863119860119861) 120582119894119895 = 623119870120582Reminus03119894119895 Re119894119895 = 120588119881119894119895119889119890

120583119894119895 119889119881119894119895119889119911 = minus2119869V119894119895ℎsp

119889119862119887119894119895119889119911 = 2119869V119894119895

ℎsp119881119894119895 (119862119887119894119895 minus 119862119901119894119895) 119889119875119889119894119895119889119911 = minus120582119894119895 120588119889119890

11988121198941198952

(13)


min119876119891119875119891119867119905119879





(15)












0

02

04

06

08

1

12

Erro

r

times10minus5


times10minus5

0

02

04

06

08

1

12

Erro

r




6 Conclusion


Fixed hiVar hi

0

02

04

06

08

1

Leng

th o

f fini

te el

emen

t



Fixed ℎiVar ℎi

016

018

02

022

024Ve

loci

ty (m

s)








Fixed hiVar hi


30

35

40

45C

once

ntra

tion

(kg

m3)


Fixed hiVar hi

0

01

02

03

04

05

Pres

sure

(bar

)



Nomenclature








Acknowledgments



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





0

01

02

03

04Le

ngth

of fi

nite

elem

ent


Fixed ℎiVar hi


0

1

2

3

4

5

Con

trol p

rofil

e (U

)

02 04 06 08 10Time (hour)


ab

0

02

04

06

08

1

Stat

e pro

file

02 04 06 08 10Time (hour)




005

01

015

02

025

Ham

ilton

ian

02 04 06 08 10Time (hour)







min int119905119891=250

(11990912 + 1199062) 119889119905st 1198891199091119889119905 = 1199092


(12)





Permissible errorx1

x2


Erro

r

0

05

1

15times10

minus5


Permissible errorx1

x2


0

05

1

15

Erro

r

times10minus5








Fixed ℎiVar ℎi

0

01

02

03

04

Leng

th o

f fini

te el

emen

t



0

1

2

3

4

5

Con

trol p

rofil

e (U

)

05 1 15 2 250Time


x1

x2

05 1 15 2 250Time

minus10

minus5

0

5

10

Stat

e pro

file





minus60

minus50

minus40

minus30

minus20

minus10

0

10

Ham

ilton

ian

05 1 15 2 250Time






119896119888119894119895 )] + 119862119901119894119895

119896119888119894119895 = 0065Re1198941198950875Sc119894119895025 sdot 119889119890119863119860119861 Sc119894119895 = 120583119894119895

(120588119863119860119861) 120582119894119895 = 623119870120582Reminus03119894119895 Re119894119895 = 120588119881119894119895119889119890

120583119894119895 119889119881119894119895119889119911 = minus2119869V119894119895ℎsp

119889119862119887119894119895119889119911 = 2119869V119894119895

ℎsp119881119894119895 (119862119887119894119895 minus 119862119901119894119895) 119889119875119889119894119895119889119911 = minus120582119894119895 120588119889119890

11988121198941198952

(13)


min119876119891119875119891119867119905119879





(15)












0

02

04

06

08

1

12

Erro

r

times10minus5


times10minus5

0

02

04

06

08

1

12

Erro

r




6 Conclusion


Fixed hiVar hi

0

02

04

06

08

1

Leng

th o

f fini

te el

emen

t



Fixed ℎiVar ℎi

016

018

02

022

024Ve

loci

ty (m

s)








Fixed hiVar hi


30

35

40

45C

once

ntra

tion

(kg

m3)


Fixed hiVar hi

0

01

02

03

04

05

Pres

sure

(bar

)



Nomenclature








Acknowledgments



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Permissible errorx1

x2


Erro

r

0

05

1

15times10

minus5


Permissible errorx1

x2


0

05

1

15

Erro

r

times10minus5








Fixed ℎiVar ℎi

0

01

02

03

04

Leng

th o

f fini

te el

emen

t



0

1

2

3

4

5

Con

trol p

rofil

e (U

)

05 1 15 2 250Time


x1

x2

05 1 15 2 250Time

minus10

minus5

0

5

10

Stat

e pro

file





minus60

minus50

minus40

minus30

minus20

minus10

0

10

Ham

ilton

ian

05 1 15 2 250Time






119896119888119894119895 )] + 119862119901119894119895

119896119888119894119895 = 0065Re1198941198950875Sc119894119895025 sdot 119889119890119863119860119861 Sc119894119895 = 120583119894119895

(120588119863119860119861) 120582119894119895 = 623119870120582Reminus03119894119895 Re119894119895 = 120588119881119894119895119889119890

120583119894119895 119889119881119894119895119889119911 = minus2119869V119894119895ℎsp

119889119862119887119894119895119889119911 = 2119869V119894119895

ℎsp119881119894119895 (119862119887119894119895 minus 119862119901119894119895) 119889119875119889119894119895119889119911 = minus120582119894119895 120588119889119890

11988121198941198952

(13)


min119876119891119875119891119867119905119879





(15)












0

02

04

06

08

1

12

Erro

r

times10minus5


times10minus5

0

02

04

06

08

1

12

Erro

r




6 Conclusion


Fixed hiVar hi

0

02

04

06

08

1

Leng

th o

f fini

te el

emen

t



Fixed ℎiVar ℎi

016

018

02

022

024Ve

loci

ty (m

s)








Fixed hiVar hi


30

35

40

45C

once

ntra

tion

(kg

m3)


Fixed hiVar hi

0

01

02

03

04

05

Pres

sure

(bar

)



Nomenclature








Acknowledgments



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





minus60

minus50

minus40

minus30

minus20

minus10

0

10

Ham

ilton

ian

05 1 15 2 250Time






119896119888119894119895 )] + 119862119901119894119895

119896119888119894119895 = 0065Re1198941198950875Sc119894119895025 sdot 119889119890119863119860119861 Sc119894119895 = 120583119894119895

(120588119863119860119861) 120582119894119895 = 623119870120582Reminus03119894119895 Re119894119895 = 120588119881119894119895119889119890

120583119894119895 119889119881119894119895119889119911 = minus2119869V119894119895ℎsp

119889119862119887119894119895119889119911 = 2119869V119894119895

ℎsp119881119894119895 (119862119887119894119895 minus 119862119901119894119895) 119889119875119889119894119895119889119911 = minus120582119894119895 120588119889119890

11988121198941198952

(13)


min119876119891119875119891119867119905119879





(15)












0

02

04

06

08

1

12

Erro

r

times10minus5


times10minus5

0

02

04

06

08

1

12

Erro

r




6 Conclusion


Fixed hiVar hi

0

02

04

06

08

1

Leng

th o

f fini

te el

emen

t



Fixed ℎiVar ℎi

016

018

02

022

024Ve

loci

ty (m

s)








Fixed hiVar hi


30

35

40

45C

once

ntra

tion

(kg

m3)


Fixed hiVar hi

0

01

02

03

04

05

Pres

sure

(bar

)



Nomenclature








Acknowledgments



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







0

02

04

06

08

1

12

Erro

r

times10minus5


times10minus5

0

02

04

06

08

1

12

Erro

r




6 Conclusion


Fixed hiVar hi

0

02

04

06

08

1

Leng

th o

f fini

te el

emen

t



Fixed ℎiVar ℎi

016

018

02

022

024Ve

loci

ty (m

s)








Fixed hiVar hi


30

35

40

45C

once

ntra

tion

(kg

m3)


Fixed hiVar hi

0

01

02

03

04

05

Pres

sure

(bar

)



Nomenclature








Acknowledgments



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Fixed hiVar hi


30

35

40

45C

once

ntra

tion

(kg

m3)


Fixed hiVar hi

0

01

02

03

04

05

Pres

sure

(bar

)



Nomenclature








Acknowledgments



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




an improved finite element meshing strategy for dynamic...

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