4.6 POD model order reduction with space-adapted snapshots for incompressible flows 106
4.6.6 Numerical example
of equations: for y`0 =y0 −˜g0 with y0 ∈ Hdiv(Ω) and ˜gj =gj+PgV ,˜jQ˜(0) for j = 0, . . . , n, find y`1, . . . , y`n∈V` such that
y`j−yj−1`
∆tj , v
!
L2(Ω)
+c(yj`, y`j, v) +c(˜gj, y`j, v) +c(y`j,g˜j, v) +a(yj`, v)
=hfj, viH−1(Ω),H01(Ω)−c(˜gj,g˜j, v)−a(˜gj, v)−
g˜j−g˜j−1
∆tj
, v
L2(Ω)
∀v∈V`.
(4.62)
Remark 4.9. Concerning the computational complexity, we have to additionally consider the projections PgV ,˜jQ˜(0) for j = 0, . . . , n in comparison to the homogeneous case. Therefore, the solution of Problem 4.5 has to be computed n+ 1 times additionally to the projections of the homogeneous solutions yhj.
Following the velocity-ROM approach b) in Section 4.6.3, we first introduce a set of modified ho-mogeneous solutions ˆyj−PV ,˜ Q˜
gj (0), forj= 1, . . . , n. These modified snapshots can be constructed using e.g. a Lagrange interpolation of the original homogeneous solutions yjh onto the reference space ˜V and an approximation of PgVjj,Qj(0) for j = 1, . . . , n. From these modified snapshots, we compute a POD basis ˆψ1, . . . ,ψˆ`y ∈ V˜. Note that these modes are in general not discrete divergence-free. Thus, they are then projected onto the space ˜Vdiv by solving Problem 4.6 with g= 0. This leads to a divergence-free velocity POD spaceV` = span(ψ1, . . . , ψ`y)⊂V˜div. Replac-ing (H01(Ω), L20(Ω)) by the pair (V`,Q) in (4.61) leads to a reduced-order model of the form (4.62).˜ Velocity-pressure reduced-order model for the inhomogeneous setting
To derive a velocity-pressure reduced-order model of the homogenized problem (4.60), we require a suitable inf-sup stable pair of reduced spaces. Since the homogenization does not alter the bilinear form b(·,·), the inf-sup stability criterion stays the same. Therefore, we compute a pressure reduced spaceQ`and a velocity reduced spaceV`like in Section 4.6.4, but using Lagrange interpolated velocity and pressure snapshots of (4.60) instead of (4.48). We derive a stable POD-Galerkin model from the time-discrete problem (4.59) by using the pair (V`, Q`) as test and trial spaces.
We solve the reduced-order model for the POD approximations y`1, . . . , yn` of the homogeneous velocity fields and the POD approximations p1`, . . . , pn` of the pressure fields. Finally, yj` +gj is a time-discrete reduced-order approximation of the velocity solution of the inhomogeneous problem.
4.6 POD model order reduction with space-adapted snapshots for incompressible flows 119
yDt (t) =
(1−14(1−cos((0.1−t)π/0.1))2 ift∈[0,0.1),
1 ift∈[0.1,1],
yxD(x) =
1− 14(1−cos((0.1−x1)π/0.1))2 ifx2 = 1, x1 ∈[0,0.1],
1 ifx2 = 1, x1 ∈(0.1,0.9),
1− 14(1−cos((x1−0.9)π/0.1))2 ifx2 = 1, x1 ∈[0.9,1],
0 otherwise.
As body force we choosef(t, x) = 0 and consider laminar flow with a Reynolds numberRe= 100.
Finite element results
The initial triangulationThinitis chosen as a criss-cross triangulation with 9 node points in both x1- andx2-directions, see Figure 48 (left). For the time discretization, we utilize a uniform time grid with time step size ∆t = 0.01 leading to n = 100 time points. In the h-adaptive finite element Algorithm 1 we choose as stopping toleranceε= 0.01 and as D¨orfler marking parameter θ= 0.1.
The resulting adaptive finite element velocity components and pressure solutions as well as the adapted meshes are shown in Figure 49 for three different time instances. The maximal number of node points in the adapted meshes is 3287. The overlay mesh (shown in Figure 48, right) has 3729 node points. For comparison, an uniform discretization with fineness as small as the smallest triangle in the adapted meshes would have 131585 node points, which leads to an out of memory error with our computing machines.
Figure 48: Initial triangulation Thinit (left) and overlay of all adapted meshes (right) POD reduced-order solutions
In order to compute a POD basis, we choose in (2.7) for the velocity POD the space X=H01(Ω) and for the pressure POD the space X = L2(Ω). As time weights, we choose αj = ∆t, i.e.
we interpret the sum in (2.7) as a quadrature of the time integral in (2.5) with a right-sided rectangle quadrature rule. This complies with the interpretation of the implicit Euler scheme as a discontinuous Galerkin method.
For the methodologies described in Sections 4.6.3 and 4.6.4, we choose the reference velocity and pressure spaces ( ˜V ,Q) as the Taylor–Hood finite element pair associated with the overlay of the˜ adapted finite element meshes, which is shown in Figure 48 (right). In Problem 4.5, we choose X =H01(Ω) in order to enable a consistent setting.
The first three velocity POD modes (without divergence-free projection) as well as the pressure POD modes and associated supremizer functions are displayed in Figure 50. Note that the POD modes computed from weakly divergence-free projected snapshots as well as the projected POD modes look very similar to the POD modes in Figure 50.
t6 t20 t100
mesh
y1,h
y2,h
ph
Figure 49: Adapted finite element meshes, velocity components yh = (y1,h, y2,h) and pressure solutions at time instances tn forn= 6,20,100
4.6 POD model order reduction with space-adapted snapshots for incompressible flows 121
i= 1 i= 2 i= 3
ψ1,i
ψ2,i
φi
Tφi
Figure 50: First three POD modes i= 1,2,3 for the velocityψi = (ψ1,i, ψ2,i) and the pressure φi, first three supremizer functionsTφi
We compare the approaches of a velocity reduced-order modeling and the velocity-pressure reduced-order modeling based on supremizer stabilization concerning accuracy and efficiency.
In the velocity-pressure approach we choose for simplicity the same number of velocity POD basis modes as pressure POD basis functions. A short-hand notation for the considered reduced-order approximations is given in Table 7.
In order to validate the approximation quality of the POD reduced-order solution we consider the following relative error for the velocity
errF E−P OD = kyh−y`kL2(0,T;Ω)
kyhkL2(0,T;Ω)
.
The results are shown in Figure 51. First of all, we observe that a velocity-pressure reduced-order model without supremizer stabilization leads to an unstable model and unsatisfactory approximation results. A naive approach, i.e. a velocity reduced-order model without utilizing a divergence-free projection leads to better approximation results in this example. From `y = 1
0 5 10 15 20 25 30 10-3
10-2 10-1 100 101
unstable ROM naive ROM div-free ROM (a) div-free ROM (b) stabilized ROM
Figure 51: Relative velocity errors of different reduced-order approximations according to Table 7 and different numbers of utilized velocity POD basis functions
method description
unstable ROM velocity-pressure reduced-order (4.50), but no supremizers
naive ROM velocity reduced-order model (4.51), but using Lagrange interpolations instead of divergence-free projections of the FE solutions
div-free ROM (a) velocity reduced-order model according to approach a) in Section 4.6.3 div-free ROM (b) velocity reduced-order model according to approach b) in Section 4.6.3 stabilized ROM velocity-pressure reduced-order model according to Section 4.6.4
Table 7: Short-hand notation for the different reduced-order models
to `y = 8, the relative error between the finite element solution and the naive reduced-order approximation decreases. However, for`y >8, the error starts to increase, which is not what we would like to achieve.
Now, let us compare the velocity reduced-order approaches of Section 4.6.3 with the velocity-pressure reduced-order approach of Section 4.6.4. All of these methodologies lead to stable results in the sense that the error between the POD reduced-order model and the finite element solution decreases for an increasing number of utilized POD modes up to `y = 8. For `y >8, the error stagnates. Both velocity approaches (a) and (b) lead to very similar results. In comparison to the velocity model, the accuracy of the velocity-pressure model is slightly better.
div-free ROM (b) stabilized ROM
FE solution 125.66 125.66
offline times
→ construction of overlay mesh 3.81 3.81
→ velocity POD 1.73 1.73
→ pressure POD – 0.05
→ div-free projection of POD modes 5.54 –
→ supremizers – 0.31
ROM solution 0.009 0.03
Table 8: Computation times (sec) for finite element solution, offline and online times using`y = 8 POD modes
4.7 POD model order reduction with space-adapted snapshots in optimal control 123
Concerning the computational efficiency, we compare the calculation times of the velocity order approach (b) with the velocity-pressure order approach. The velocity reduced-order approach (a) is more expensive than the approach (b), since Problem 4.5 has to be solved for each snapshot, i.e. ntimes, whereas in the latter approach it only has to be solved for each velocity POD mode, i.e. `y < n times. For comparison, we list the computational time for the adaptive finite element solution in Table 8. It turns out that the divergence-free projection of the POD modes is more expensive than the computation of the supremizer functions. Concerning the reduced-order online time, the solution of the velocity model is very fast with only 0.009 seconds which leads to a speed up of factor 13962 compared to the adaptive full-order finite element simulation. The solution of the velocity-pressure reduced order model takes 0.03 sec-onds, so it is slower than the velocity reduced-order model. This is due to the fact that the dimension of the velocity model depends solely on `y, whereas the complexity of the velocity-pressure reduced-order model additionally depends on the number of supremizer functions and the number of pressure POD basis functions. Recall that in this test run, we take for simplicity the same number of POD modes for the pressure as for the velocity, i.e.`p =`y.
These factors motivate to utilize POD reduced-order modeling in a multi-query setting. We note that a finite element simulation on a grid with the same fineness as the smallest triangle as in the adapted meshes is not possible on our computing machines since it reaches the memory limit.