The Application of a Black-Box Solver with Error Estimate to Different Systems of PDEs

Transcription

1 The Application of a Black-Bo Solver with Error Estimate to Different Sstems of PDEs Torsten Adolph and Willi Schönauer Institut für Wissenschaftliches Rechnen, Forschungszentrum Karlsruhe torstenadolph@iwrfzkde willischoenauer@iwrfzkde Happ are those people that do not see the errors Summar At first a brief overview of the Finite Difference Element Method (FDEM) is given, above all how an eplicit estimate of the error is obtained Then for some academic eamples the estimated and eact error are compared showing the qualit of the estimate The PDEs for fuel cells of PEMFC and SOFC tpe with etremel nonlinear coefficients are solved and the error estimate shows the qualit of the solution Finall for a complicated fluid/structure interaction problem of a high pressure Diesel injection pump, where the domain of solution has 3 subdomains with different PDEs and where a nested iteration procedure is needed, the PDEs are solved and the global error estimate shows the qualit of the solution For all these eamples it would be ver difficult to obtain a qualit control of the solution b conventional grid refinement tests 1 Introduction The development of the Finite Difference Element Method (FDEM) at the computer center of the Universit of Karlsruhe has been supported b the German Ministr of Research (BMBF) The application of FDEM to the numerical simulation of fuel cells (FCs) has been supported b the Research Alliance Fuel Cells of the state Baden-Württemberg In this paper we present a compilation of results of these projects Never before such problems have been solved with error estimates So the emphasis of this paper will be on the error estimate: together with the solution we present values or plots for the error estimates Because of the limited accorded space of the paper we cannot present all the details of FDEM and of the eamples However, we will give the precise information where these details are in the corresponding reports As these reports are in the Internet, the reader can immediatel have a look at them at his computer

2 2 Torsten Adolph and Willi Schönauer 2 The Finite Difference Element Method (FDEM) FDEM is an unprecedented generalization of the FDM on an unstructured FEM mesh It is a black-bo solver for arbitrar nonlinear sstems of 2-D and 3-D elliptic or parabolic PDEs If the unknown solution is u(t,,, z) the operator for PDEs and BCs (boundar conditions) is (241) and (242) in [1]: Pu P (t,,, z, u, u t,u,u,u z,u,u,u zz,u,u z,u z )=0 (1) For a sstem of m PDEs u and Pu have m components: u 1 u =, Pu = u m P 1 u P m u (2) Because we have a black-bo solver, the PDEs and BCs and their Jacobian matrices of tpe (246) in [1] must be entered as Fortran code in prescribed frames The geometr of the domain of solution is entered as a FEM mesh with triangles in 2-D and tetrahedra in 3-D The domain ma be composed of subdomains with different PDEs and non-matching grid From the element list and its inverted list we determine for each node more than the necessar number of nodes for difference formulas of a given consistenc order q B a sophisticated algorithm from this set the necessar number of nodes is selected, see Section 22 in [1] From the difference of formulas of different consistenc order we get an estimate of the discretization error If we want eg the discretization error for u and u,d,q denotes the difference formula of consistenc order q, the error estimate d is defined b d := u,d,q+2 u,d,q, (3) ie b the difference to the order q + 2 This has a built-in self-control: if this is not a better formula the error estimate shows large error With such an error estimate we can eplicitl compute the error of the solution b the error equation (248) in [1] The knowledge of the error estimate allows a mesh refinement and order control in space and time (for parabolic PDEs), see Section 25 in [1] A special problem for a black-bo solver is the efficient parallelization because the user enters his domain b the FEM mesh We use a 1-D domain decomposition with overlap to distribute the data to the processors, see Section 28 in [1] We use MPI A detailed report on the parallelization is [2] The resulting large and sparse linear sstem is solved b the LINSOL program package [3] that is also efficientl parallelized for iterative methods of CG tpe and (I)LU preconditioning

3 3 Academic Eamples PDE Black-bo Solver with Error Estimate 3 The purpose of these academic eamples is to check the qualit of the error estimate (which is one level higher than the usual check for the qualit of the solution), see Section 210 in [1] We define the global relative error for a component l of the solution and the global relative error b u d,l u d,l, u d u d =ma u d,l l u d,l, (4) where u d,l is computed from the error equation (component l of (248) in [1]) The norm is the maimum norm We generate from the original PDE Pu = 0 a test PDE for a given solution ū b Pu Pū =0thathasū as solution This prescription holds also for the BCs The eact global relative error then is ū u d (5) u d We compute on the HP XC6000 with Itanium2 processors of 15 GHz (Universit of Karlsruhe) As eact solution ū we select either a polnomial of a given order or a sugar loaf tpe function (21016) in [1] We solve the Navier-Stokes equations in velocit/vorticit form (21013) in [1] with the unknown functions velocit components u, v and vorticit ω, and Renolds number Re =1 We solve on a circle with radius = 1 on a grid with 751 nodes, 1410 elements that has been generated b the commercial mesh generator I-DEAS We compute with 8 processors The given CPU time is that of the master processor 1 Table 1 Results for the solution of the Navier-Stokes tpe equations on a circle with 751 nodes for different consistenc orders q and test function ū order q =2 orderq =4 orderq =6 error eact CPU error eact CPU error eact CPU tpe ū error estim sec error estim sec error estim sec pol E E order E E-08 sugar 0694E E E loaf 0642E E here the order 8 for the error estimate is overdrawn (too coarse grid) Table 1 shows the results Here are two remarks: for ū polnomial of order 6 and consistenc order q = 6 we should reproduce ū eactl which is epressed b the small errors For the sugar loaf function and consistenc order q =6 we get a large error estimate This shows the built-in self-control: near the top of the sugar loaf the grid is too coarse for the consistenc order q +2=8

4 4 Torsten Adolph and Willi Schönauer that is used for the error estimate, the order 8 is overdrawn (higher order ma not be better) Table 2 Results for the self-adaptation of mesh and order for sugar loaf test function for prescribed global relative error (025%) no no no of no of nodes global relat sec of of nodes with order error for ccle nodes elem ref eact estimated ccle E E E E E E For the demonstration of the self-adaptation we solve the same problem with the sugar loaf function again with 8 processors, but now we switch on the mesh refinement and order control for a global relative error of 025% The results are shown in Table 2 The requested accurac needs 3 refinement ccles no of nodes ref is the number of refinement nodes that determine the refinement elements from which then follows the new number of nodes Observe the ecellent error estimate that results from the optimal local order Figure 1 shows the grid after the 3 rd ccle, the refinement is clearl visible 1 rd Grid of 3 ccle Fig 1 Refined grid after 3 rd ccle of Table 2 As mentioned above we wanted to demonstrate the qualit of the error estimate The estimate is the better the smaller the error is, this is a natural

5 PDE Black-bo Solver with Error Estimate 5 consequence of (3) What we have seen in Table 1 is also part of our test technique for each new problem: we at first create from the new problem a test PDE Pu P ū = 0 and check with polnomial test solutions ū the error estimate like in Table 1 4 The Numerical Simulation of Fuel Cells (FCs) This project was supported b the Research Alliance Fuel Cells (FABZ) of the state of Baden-Württemberg The corresponding report [4] was written b ZSW Ulm for the equations of the PEMFCs (Part I), b the IWE of the Universit of Karlsruhe for the equations of the SOFCs (Part II) and b the RZ of the Universit of Karlsruhe for the numerical solution of the corresponding PDEs (Part III) These problems are characterized b the etreme nonlinearit of their coefficients that depend in a complicated wa on the variables themselves Quite naturall we will report here onl on that part of [4] that deals with the numerical solution of the PDEs Up to now nobod had solved these PDEs with an error estimate 41 Numerical Solution of the PDEs for PEMFCs The PEMFCs (Polmer Electrolte Membrane FC or Proton Echange Membrane FC) are the cold FCs, operating at about 300 K The domain of solution for the used model is the GDL (gas diffusion laer) that is at its lower left half open to the ogen channel and at its lower right half closed b a rib, see Fig 2 How this is cut out of a whole cell can be seen at Fig 1 on page III4 in [4] t GDL (019 mm) smmetr line reaction laer c/2+r/2 (1 mm) GDL smmetr line c/2 r/2 (05 mm) (05 mm) channel rib Fig 2 Domain of solution for the PEMFC The variables for this problem are the molar flu densities of ogen in -direction ṅ o and -direction ṅ o, similarl for water vapor ṅ w and ṅ w,and

6 6 Torsten Adolph and Willi Schönauer for nitrogen ṅ n and ṅ n, the partial pressure for ogen p o,forwaterp w and for nitrogen p n, the total pressure p and as a special variable that has phsical meaning onl at the reaction laer the current densit i So we have 11 variables and need a sstem of 11 PDEs These PDEs are given in Part I of [4], and in Table 1 on page III6 is shown which equation is used for which variable position We show below as an eample a tpical species transport equation, but without eplanation of the notations: ṅ o DKn o + pwṅ o poṅ w D owp + pnṅ o poṅ n D onp + 1 RT [ + Bo DKn o + Bopw D owp ( ) 1 Bw B o + Bopn D onp p o po RT p ( 1 Bn B o )] po p p RT p =0 (6) The effective permeabilities B j depend in a complicated nonlinear wa from the pressure p and the partial pressures p i We do not discuss here the BCs, the are given on pages III9 III11 in [4] n-dot-w error-n-dot-w- 55E-02 50E-02 45E-02 40E-02 35E-02 30E-02 25E-02 20E-02 15E-02 10E-02 50E Fig 3 Contour plot of molar flu densit of water vapor in -direction ṅ w and its error

7 PDE Black-bo Solver with Error Estimate 7 We computed with 32 processors of the HP XC6000 with 15 GHz Itanium2 processors on a grid of nodes in,-direction, we used consistenc order q = 4 Because we have 11 variables per node we have unknowns The computation time was 4123 sec on the master processor 1 Figure 3 shows a tpical result for ṅ w and its error There is a quasisingularit at the lower boundar where the BCs change from channel to rib Figure 4 shows the current densit i and its error along the reaction laer This is the most interesting quantit for the engineers i mean = 2981,92 i ,0000 0,0002 0,0004 0,0006 0,0008 0,0010 error-i 2,0E-02 1,6E-02 1,2E-02 8,0E-03 4,0E-03 0,0E+00 0,0000 0,0002 0,0004 0,0006 0,0008 0,0010 Fig 4 Current densit i along the reaction laer and its error In [4] on pages III15 III24 are figures of tpe Fig 3 for all variables So the engineer can see if he can trust the solution If there would be no error estimate he had to do grid refinement tests and observe for all variables and nodes (here values) how the change with finer grid So the error estimate that consumes onl a fraction of the total computation time is an invaluable feature in the solution process

8 8 Torsten Adolph and Willi Schönauer 42 Numerical Solution of the PDEs for SOFCs The SOFCs (Solid Oide FCs) are the hot FCs, operating at about 1200 K The domain of solution for the used model is the anode with flow in porous media and the gas channel with Navier-Stokes equations, see Fig 5 electrolte 1 2 anode side 2 DL side 1 gas channel 6 gas flow d A d K rib l K Fig 5 Domain of solution for the SOFC and numbering of the boundaries Here we have a solution domain that consists of two subdomains with different PDEs and a dividing line in-between for which we must prescribe coupling conditions (CCs) which are interior BCs Because the model includes methane reforming, the variables are the flow velocities u, u in - anddirection, the mole fractions Y CH4 = Y 3 for methane, Y CO = Y 4 for carbon monoide, Y H2 = Y 5 for hdrogen, Y CO2 = Y 6 for carbon dioide, Y H2O = Y 7 for steam, and pressure p Sowehave8variablesandneed8PDEsforthe anode domain and 8 PDEs for the gas channel domain These PDEs are given in Part II of [4] In Table 12 on page III31 the sequence of the variables and equations in the channel is given and in Table 15 on page III36 in the anode In the channel we have Navier-Stokes-tpe equations, eg the -momentum, (18) on page III32 in [4]: ( µ u ϱu ϱu u + p µ ) 2 u u u 2 ( µ 4 u ( u ) u + u ) =0, and we have transport equations like that for methane, (19) on page III32 in [4]: (7) p u Y 3 u py 3 Y3 pu p u Y 3 u py 3 Y3 ( ) ( p Y 3 + Y3 p + D 2 p 3,gas Y p Y D3,gas pu + 2 Y 3 ) p =0 (8) In the anode the Navier-Stokes equations are replaced b Darc s law The species transport equations have additional terms b the -dependence of p

9 PDE Black-bo Solver with Error Estimate 9 and b the chemical reactions The transport equation for methane, (34) on page III37 is now + D3,gas p u Y 3 u py 3 Y3 pu p u Y 3 u py 3 ( Y3 pu + D 2 p 3,gas ( ) ( p Y 3 + Y3 p + D 2 p 3,gas + D3,gas Y p ) Y 3 + p 2 Y 3 2 ) Y p Y 3 + p 2 Y 3 ( ) 2 p Y 3 + Y3 p RT d A r 3 =0 These equations are etremel nonlinear because µ in (7) and the D i,gas depend nonlinearl on the Y j, similarl the reaction rates r k are depending Here we must define the BCs for the 6 boundaries of Fig 5 and the CCs for side 1 and side 2 of the dividing line DL These conditions are given in Tables on pages III40 and III41 in [4] (9) Anode Y-CO E-01 16E-01 14E-01 12E-01 10E-01 80E-02 60E-02 40E-02 20E Anode error-y-co E-02 35E-02 30E-02 25E-02 20E-02 15E-02 10E-02 50E-03 Fig 6 Contour plot of mole fraction Y CO and its error for the anode The numerical solution of these equations was much more critical than that of the PEMFCs because of the etreme nonlinearit of the coefficients

10 10 Torsten Adolph and Willi Schönauer We computed with a grid with 80 nodes in -direction and 41 nodes in the channel and 41 nodes in the anode in -direction, resulting in 52480unknowns We used consistenc order q = 4 The computation time on 8 processors of the HP XC6000 with processors Itanium2, 15 GHz, was 510 sec on the master processor 1 The colour plots of the results and error estimates for all 8 variables in the channel and anode are presented in [4], pages III47 III62 We present here onl the mole fraction Y CO and its error in Fig 6 for the anode The ampleness of the generated information can be estimated onl if one looks at all the nice colour plots of the report [4] Here the engineer can immediatel see the qualit of the solution from the error plots, this is an unprecedented gain in information 5 Fluid/Structure Interaction for a High Pressure Diesel Injection Pump A detailed presentation of this problem (cooperation with Bosch) is given in Section 33 in [1] In a high pressure Diesel injection pump the housing etends under the injection pressure of 2000 bar and the piston is compressed so that the lubrication gap between housing and piston changes its form and consequentl the leakage flow changes This is a fluid structure interaction problem The problem is simplified b replacing the complicated shape of the housing b a tube or bush, see Fig 7 The piston does not move, so we have a static configuration piston gap housing z Fig 7 Smbolic configuration and dimensions in mm In realit the gap is etremel thin Our domain of solution now has 3 subdomains with different PDEs: in the housing and piston we must solve the elasticit equations of steel, in-between we have the gap with the Navier-Stokes equations for Diesel The coupling between these domains is the following: the fluid pressure p is the normal stress for housing and piston, so we have a direct interaction of the flow on

11 PDE Black-bo Solver with Error Estimate 11 the structure B this normal stress the housing epands and the piston is compressed, thus the form of the gap changes and this changes the flow which changes p and thus the normal stress etc So the interaction of housing and piston on the flow is indirect and more complicated and requires an iterative procedure We solve the problem in aismmetrical coordinates, then in Fig 7 becomes the radius r For the elasticit equations in housing and piston the dependent variables are the displacements w and u in z- andr-direction, the stresses σ z, σ r, σ ϕ and the shear stress τ rz (= τ zr ) Although we have rotational smmetr with / ϕ = 0, there is circumferential stress σ ϕ So we have 6 variables and need 6 PDEs The are given in (3341) (3346) on page 129 in [1] Here we show the first and last of these equations in incremental form (inde old is for the last solution): 1 E [σ z σ z,old ν(σ ϕ σ ϕ,old ) ν(σ r σ r,old )] w =0, (10) z τ rz r + σ z z + τ rz r =0 (11) Here E is the elasticit module and ν is Poisson s ratio Table (3347) in [1] gives the information which equation is used in which position (for which variable) in the sstem of 6 equations In the lubrication gap we must solve the Navier-Stokes equations The variables are the velocit components w and u in z- andr-direction, and the pressure p So we need a sstem of 3 equations Here we show the momentum equation in r-direction and the continuit equation, (33417) and (33419) on page 123 in [1]: u u r + w u z + 1 p ϱ r η ϱ ( 2 u r u r r u ) r u z 2 =0, (12) u r + u r + w =0 (13) z Here η and ϱ are the dnamical viscosit and the densit But now we have a problem: the whole domain has 3 subdomains where two of them have 6 variables and PDEs, and one has 3 variables and PDEs However, the code is designed for the same number of variables in the whole domain So we add in the fluid domain 3 dumm variables with variable = 0 as PDEs and BCs Table (33420) on page 133 in [1] shows which equation is used in which position of the sstem, ie for which preferred variable: for w we take the continuit equation, for u the r-momentum equation and for p the z-momentum equation This is a quite natural ordering If we want to discuss the BCs for the flow we immediatel meet a serious problem: at the entr we have a prescribed pressure of 2000 bar = 200 N/mm 2 and at the eit of 0 bar Because in the Navier-Stokes equations there is onl p/ z we can prescribe for incompressible flow the pressure onl at one position, eg 2000 bar at the entr Then the

12 12 Torsten Adolph and Willi Schönauer pressure at the eit is the result of the Navier-Stokes equations At the high pressure entr we prescribe a parabolic velocit profile for w: w(r) =parabola with w ma in the middle of the entr The choice of w ma determines the pressure at the eit, ie the pressure drop in the gap This introduces what we call w ma -iteration: we start with an appropriate value of w ma, determine p eit and if it is too large we increase w ma (gives larger pressure drop), if it is too small, we reduce w ma This is made b a sophisticated iteration procedure The detailed discussion of the BCs for housing, piston and fluid is given in Section 334 in [1] and would be too length for this paper However, the essential part is the coupling between fluid and structure: if we have computed displacements for the pressure p given b the fluid flow, we appl in an incremental form these displacements b a shift algorithm that is presented in Section 333 in [1] The inde old in (10) denotes the values before the shifting (on the old grid) Then we observe if the grid still shifts This induces what we call grid iteration If the grid does no longer move we have the solution of our problem Thus we have the nested iterations shown in Fig 8: pressure incrementation grid iteration w ma-iteration Newton iteration Fig 8 The nested iterations for the solution process The innermost iteration is the Newton iteration for the solution of the PDEs, then we must determine w ma for the eit pressure zero, then we must appl the displacements until the grid does no longer move The outermost iteration gives the possibilit to increase graduall the entr pressure When we made the first numerical eperiments we had serious difficulties caused b the etreme differences in length scales: housing and piston in cm, lubrication gap in micrometer, see Fig 7 After some trials we decided to use cm as length scale The discretization errors of housing and piston for the used grid caused the surfaces to be rough in the micrometer scale Therefore we applied a smoothing of these surfaces We used a grid of in z,r-direction for the housing, for the fluid and for the piston We computed with 16 processors of the HP XC6000 with 15 GHz Itanium2 processors The CPU time on the master processor 1 was 3354 sec, of which 3296 sec were needed for the linear solver LINSOL with full LU preconditioning Table 3 gives some results for 2000 bar entr pressure for housing and fluid In Table 3351 on page 138 of [1] further values for entr pressures of 1500 bar to 3000 bar are given Figure 9 shows the

13 PDE Black-bo Solver with Error Estimate 13 Table 3 Maimum value of solution component, ma relat error and mean relat error of solution component and volume flow through the gap for entr pressure of 2000 bar Housing ma solution ma error mean error w cm 04143E E E 04 u cm 07584E E E 04 σ z N/cm E E E 04 σ r N/cm E E E 04 σ ϕ N/cm E E E 04 τ rz N/cm E E E 04 Fluid ma solution ma error mean error Volume [cm 3 /s] w cm/s 03339E E E u cm/s 03810E E E+01 p N/cm E E E 02 form of the lubrication gap for 2000 bar entr pressure The bold lines show the original channel It is amazing how the high injection pressure changes the lubrication gap from the manufacturing dimension of 25 micrometer to up to 115 micrometer The error estimates in Table 3 show the qualit of the solution For the fluid there is a maimal error of w of 87%, but the (arithmetic) mean error is 14%, so large errors occur onl locall Fluid domain 2000 bar r(cm) z(cm) Fig 9 Fluid domain with computational grid for 2000 bar entr pressure and original channel (bold lines)

14 14 Torsten Adolph and Willi Scho nauer Figure 10 shows the contour plot of the velocit w in z-direction which is responsible for the leakage flow that is in this case 240 cm3 /s Figure 10 also shows its error plot Here we can see that the large errors occur onl locall There a much finer grid should be used As the mean error of w is 14% we can conclude that the volume flow is also accurate to this error level Note that this is a global error estimate that includes all the errors of all the equations in the coupled domains Here again the error estimate gives us the certaint that we can trust our solution The reader ma think how he could get this certaint b other methods for this complicated fluid/structure interaction problem w (cm/s) 2000 bar r (cm) z (cm) 3 4 error-w 2000 bar E-01 60E-01 40E-01 20E-01 10E-01 50E r (cm) z (cm) 3 4 Fig 10 Contour plot for the velocit w in z-direction for 2000 bar and its error 6 Concluding Remark The purpose of this paper was to show that an error estimate is an invaluable advantage for a PDE solver We have shown for some academic eamples

15 PDE Black-bo Solver with Error Estimate 15 where we constructed PDEs with known solution, that our error estimate gives an ecellent approimation for the eact error It is the better the smaller the error is Then we solved the PDEs for the numerical simulation of fuel cells of PEMFC and SOFC tpe and the error estimates showed the qualit of the solution for all components of the sstems Finall, we solved a fluid/structure interaction problem for a high pressure Diesel injection pump The high pressure of 2000 bar bends up the housing that the lubrication gap widens from 25 micrometer up to 115 micrometer The solution algorithm is a complicated nested iteration Nevertheless we compute a global error estimate for the coupled domains of housing, piston and fluid that tells us that we can trust our solution The conventional wa would be to do a sequence of grid refinements The reader ma imagine what amount of work would be necessar to obtain a comparable information like our error estimate References 1 W Schönauer, T Adolph, FDEM: The Evolution and Application of the Finite Difference Element Method (FDEM) Program Package for the Solution of Partial Differential Equations, 2005, available at wwwrzuni-karlsruhede/rz/docs/fdem/literatur/fdempdf 2 T Adolph, The Parallelization of the Mesh Refinement Algorithm in the Finite Difference Element Method, Doctoral Thesis, 2005, available at wwwrzuni-karlsruhede/rz/docs/fdem/literatur/par mra fdempdf 3 LINSOL, see wwwrzuni-karlsruhede/rd/linsolphp 4 The Numerical Simulation of Fuel Cells of the PEMFC and SOFC tpe with the Finite Difference Element Method (FDEM), collective paper b three groups, 2005, available at wwwrzuni-karlsruhede/rz/docs/fdem/literatur/fuelcellspdf