Numerical tsunami simulation using goal-oriented mesh adaptivity

NUMERICAL TSUNAMI SIMULATION USING GOAL-ORIENTED MESH ADAPTIVITY Nicolas Barral,1David Ham,1Matthew Piggott,1Joe Wallwork,1,2 Hilary Weller3. 1Imperial College London, 2Mathematics of Planet Earth Centre for Doctoral Training, 3University of Reading. INTRODUCTION In shallow water tsunami modelling and hazard mitigation there is a clear objective: to minimise the error accrued in approximating the free surface displacement near to coastal regions of importance. An adjoint approach enables incorporation of such regions into a mesh adaptive algorithm, providing an efficient simulation meeting the objective above. SHALLOW WATER EQUATIONS For a region of ocean Ω⊂Ò2, define the fluid velocity - free surface displacement tuple q=(u,η) : Ω→Ò3, bathymetry b:Ω→Òand viscosity νh:Ω→Ò. For gravitational acceleration g>0,            ∂u ∂t+(u·+)u+g+η+fD z×u= + ·(νh(+u+(+u)T)), ∂η ∂t+ + ·((η+b)u)=0. (1) ANISOTROPIC MESH ADAPTIVITY This mesh adaptive process is driven by Riemannian metric fields M:Ω→Òn×nin nD. At each x∈Ω, the eigenpairs of Mdescribe the transformation from Euclidean space into the space defined by M. (1,0) (0,1) M1 2 M−1 2 (h1,0) (0,h2) Å2=(Ò2,I2) (Ò2,M) Given a Ð1 metric defined on a mesh, anisotropic mesh adaptivity enables the generation of a new mesh using node insertion,node deletion,edge swapping and node movement (in the 2D case). Anisotropic mesh adaptivity is implemented in Thetis using PRAgMATIc (PaRallel Anisotropic Mesh Adaptive ToolkIt). GOAL-ORIENTED MESH ADAPTIVITY For a spatial region A⊂Ω, consider the objective functional J(q)=ZTend Tstart "A η(x,y,t)dxdydt.(2) In the case of the T¯ ohoku tsunami, consider Aas a region surrounding Fukushima’s Daiichi nuclear power plant, say. Using (2) we may formulate the associated adjoint shallow water equations. Pyadjoint enables the generation of adjoint solutions in a “discretise-then-differentiate” type approach [Gun02]. Goal-oriented mesh adaptivity seeks a mesh upon which we minimise error in evaluation of (2): min H`J(q)−J(qh)`.(3) Having solved (1) on a relatively coarse mesh, adjoint solutions are backed out by pyadjoint. Various a posteriori error estimators may be constructed, such as the dual-weighted residual (DWR). On each element DWR gives error indicators EK h= ¥Rh(qh),λh¦L2(K),K∈ H .(4) Here Rhdenotes the (strong) residual formed by interpreting (1) as a 3-component vector functional equal to zero and (qh,λh)form our primal-dual approximation tuple. In the final run, the indicators (4) are used to construct a metric by Ð1 interpolation and scaling of the identity matrix, using which (1) is solved adaptively. CODE In order to run a mesh adaptive DWR simulation for any realistic or idealistic tsunami, all the user need do is specify the initial mesh, bathymetry, initial and boundary conditions, Coriolis coefficient of choice and any parameters which differ from the default settings. Figure: A sample script to run a mesh adaptive DWR simulation of the T¯ ohoku tsunami. RESULTS Quality of an approximation is measured using relative error in objective functional evaluation. An ‘exact’ value for (2) is obtained using a high resolution fixed mesh run. Figure: Initial condition blank space bl[SI11]. Figure: 50km radius of importance surrounding Fukushima. Figure: Mesh adapted using DWR indicators. CONCLUSIONS IThe adjoint problem allows guidance of the mesh adaptive process, offering additional a posteriori information. This is particularly useful if only a small number of regions are of interest or importance. IWhilst error and element count may be reduced, there is progress to be made in reducing CPU time. FURTHER RESEARCH IConsider mesh movement in the context of tsunami modelling. IFormulate a hybrid hr-adaptive method. IInvestigate mesh adaptivity in other ocean modelling applications, such as storm surges and Gulf Stream separation. REFERENCES Gunzburger (2002), Perspectives in flow control and optimization. Saito et al. (2011), Tsunami source of the 2011 T¯ ohoko-Oki earthquake, Japan: inversion analysis based on dispersive tsunami simulations. GitHub github.com/jwallwork23 Firedrake firedrakeproject.org Thetis thetisproject.org This work could not have been done without the generous support of