Goal-oriented error estimation in tidal turbine modelling

GOAL-ORIENTED ERROR ESTIMATION IN TIDAL TURBINE MODELLING Joe Wallwork,1,2 Matthew Piggott,1Nicolas Barral,1Stephan Kramer,1David Ham,3Hilary Weller4. 1Earth Science & Engineering, ICL, 2MPE CDT, 3Mathematics, ICL, 4Meteorology, University of Reading. ABSTRACT The location of new tidal power plants impacts highly upon their effectiveness. The positioning of a tidal array may be framed mathematically as an optimisation problem, whose solution can lead to an increase in yield [Funke, Farrell, and Piggott, 2014]. The objective functional (relating to the power output) used in the optimisation problem also provides a basis for goal-oriented error estimation. By application of dual weighted residual error estimation to the nonlinear shallow water equations, we have developed a mesh optimisation algorithm which seeks to minimise the error in the assessment of tidal power plant output. A discontinuous Galerkin approach is applied, using the Thetis coastal ocean finite element model [Kärnä et al., 2018]. In this preliminary work, mesh optimisation is performed for steady simulations. TIDAL TURBINE PROBLEM Consider an idealised channel Ω⊂Ò2, with inflow and outflow boundaries Γ 1and Γ 2, as shown in Figure 1. Two turbines T={T1,T2} are positioned in the centre of the channel. The stationary, nonlinear shallow water equations, (u·+)u+g+η+Ct η+bkuku= + · (ν+u),+· ((η+b)u)=0,(1) model flow in the channel, through the fluid velocity - free surface elevation tuple, q=(u,η). As well as bathymetry band viscosity ν, we consider a drag coefficient, Ct(x)=Cb+1 2Õ T∈T ρCTAT1T(x),(2) for background drag Cband turbine density ρ. Associated with turbine T are the thrust coefficient CT, area ATand footprint indicated by 1T. Given the shallow water equations (1), solved for qon some mesh Ωh, we seek to accurately approximate the turbine array power output, J(q)=Õ T∈T ∫T Ctkuk3dx.(3) SOLUTION STRATEGY We solve (1) using the discontinuous finite element space Ö:= Ð1DG ×Ð1DG and the symmetric interior penalty method described in [Epshteyn and Rivière, 2007]. This involves integration by parts in all but the drag term, meaning the variational problem BDG(q,ψ)=`DG(ψ),[ψ∈Ö(4) includes inter-element flux terms. Define R(q,·) :=`DG(·) − BDG(q,·). Having solved the forward problem in weak form (4) with Thetis, the high level symbolic differentiation tool pyadjoint transforms operations in the forward computation appropriately, so as to automatically generate and solve the corresponding adjoint problem, ∂R ∂q T λ=∂J ∂q T .(5) GOAL-ORIENTED ERROR ESTIMATION Given finite element approximations qhand λhof the forward and adjoint problems (1) and (5), it is shown in [Becker and Rannacher, 2001] that J(q) − J(qh)=R(qh,λ−λh)(DWR)+R(2)(6) where R(2)is quadratic in the forward and adjoint errors. As such, (DWR)provides a first order approximation to the error in (3). We obtain elementwise error indicators by considering the absolute value of (DWR) on each K∈Ωh. NUMERICAL EXPERIMENTS The local error indicators may be used as a basis for mesh adaptivity; the mesh is refined wherever their values are significant and coarsened wherever they are sufficiently small. Γ 1Γ 2 T1T2 Figure 1: Domain setup, illustrated on initial mesh Figure 2: Fluid speed, as computed on initial mesh. Figure 3: Adjoint fluid speed, as computed on initial mesh. Figure 4: Mesh adapted with respect to vorticity, ω=b z· (+×u). Figure 5: Mesh adapted with respect to (DWR). Intuitively, the finest mesh resolution should be near to and inbetween the two turbines, with coarse resolution acceptable downstream. The preliminary results above indicate that the DWR error estimation approach leads to such meshes. This is due to (6), which states how the DWR estimator prioritises error accrued in objective functional computation as the most important error quantity. A mesh adapted with respect to vorticity, on the other hand, has no knowledge of (3), using more downstream resolution than necessary. FURTHER RESEARCH IConsider time dependent case with tides. IConsider a realistic problem with more turbines. REFERENCES Becker, Roland and Rolf Rannacher (2001). “An optimal control approach to a posteriori error estimation in finite element methods”. In: Acta numerica 10. Epshteyn, Yekaterina and Béatrice Rivière (2007). “Estimation of Penalty Parameters for Symmetric Interior Penalty Galerkin Methods”. In: J. Comp. Appl. Math. 206.2. Funke, Simon W, Patrick E Farrell, and MD Piggott (2014). “Tidal turbine array optimisation using the adjoint approach”. In: Renewable Energy 63, pp. 658–673. Kärnä, Tuomas et al. (2018). “Thetis coastal ocean model: discontinuous Galerkin discretization for the three-dimensional hydrostatic equations”. In: Geosci. Model Dev. 11, pp. 4359–4382. GitHub github.com/jwallwork23 Firedrake firedrakeproject.org Thetis thetisproject.org This work could not have been done without the generous support of