15:45
Time Integration and Spatial Discretisations (Part 3)
Chair: Florin Adrian Radu
15:45
25 mins
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A practical a posteriori strategy to determine the optimal number of degrees of freedom for hp-refinement in finite element methods
Jie Liu, Henk M. Schuttelaars, Matthias M\"oller
Abstract: A common practice in improving the accuracy of finite element solutions is to use h-, p- and hp- refinement strategies. When using h-refinement, while keeping the approximation order of the element, p, fixed, the aim is to decrease the truncation error by increasing the number of degrees of freedom (''DoFs'').
However, when the number of DoFs becomes larger than a critical number N(p)_opt, as a function of p, computational round-off errors accumulate, and start to exceed the truncation error. Since further refinements will even result in less accurate solutions, the total discretization error (truncation + round-off error) Err(p)_min corresponding to N(p)_opt is the minimum attainable error for p. Focusing on one-dimensional differential equations, we investigate this phenomenon and propose a systematic approach to identify N(p)_opt a posteriori, both for the primary variable and its derivatives. We show that both N(p)_opt and Err(p)_min decrease for increasing p, which has led us to develop a practical a posteriori hp-refinement strategy that adjusts the mesh width h(p) in accordance with p so that for each p the optimal mesh width h(p)_opt correlates with N(p)_opt.
Furthermore, we show that the mixed FEM incurs smaller round-off errors compared with the standard FEM, thus allowing for a more accurate solution and also its derivatives.
Moreover, the possible influence factors on the discretization error, such as the L2 norm of the solution, working precision, computational mesh, type and implementation of boundary conditions and choice of solver, are also investigated. Finally, our strategy is successfully applied on a Helmholtz equation.
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16:10
25 mins
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A residual-based artificial viscosity finite difference method for scalar conservation laws
Lukas Lundgren, Vidar Stiernström, Murtazo Nazarov, Ken Mattsson
Abstract: We present a residual-based artificial viscosity finite difference method for solving scalar conservation laws. The artificial viscosity is constructed such that at most first-order upwind dissipation is applied at shocks, without destroying accuracy in smooth regions. The spatial discretization uses summation-by-parts (SBP) finite difference operators and boundary conditions are imposed weakly through simultaneous approximation terms (SAT), allowing for fully explicit time integration of the resulting scheme. The method is benchmarked against the 2D advection equation and the Kurganov-Petrova-Popov (KPP) rotating wave problem. In Figures 1 and 2 computed solutions are presented using 401x401 grid points and 5th order accurate spatial discretizations.
Furthermore convergence for smooth problems has been verified, where the solution converges with the accuracy of the spatial discretization. This has been tested using 3rd up to 9th order accurate SBP finite difference operators.
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16:35
25 mins
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Adaptive Numerical Simulation of a Phase-field Fracture Model in Mixed Form tested on an L-shaped Specimen with High Poisson Ratios
Katrin Mang, Mirjam Walloth, Thomas Wick, Winnifried Wollner
Abstract: This work presents a new adaptive approach for the numerical simulation of a phase-field model for fractures in nearly incompressible solids. In order to cope with locking effects, we use a recently proposed mixed form where we have a hydro-static pressure as additional unknown besides the displacement field and the phase-field variable. To fulfill the fracture irreversibility constraint, we consider a formulation as a variational inequality in the phase-field variable. For adaptive mesh refinement, we use a recently developed residual-type a posteriori error estimator for the phase-field variational inequality which is efficient and reliable, and robust with respect to the phase-field regularization parameter. The proposed model and the adaptive error-based refinement strategyare demonstrated by means of numerical tests derived from the L-shaped panel test, originally developed for concrete. Here, the Poisson's ratio is changed from the standard setting towards the incompressible
limit $\nu \to 0.5$.
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17:00
25 mins
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Stochastic Optimal Control of Renewable Energy
Renzo Caballero, Raúl Tempone
Abstract: Uruguay has always been a pioneer in the use of renewable sources of energy. Nowadays, it can usually satisfy its total demand from renewable sources, but half of the installed power, due to wind and solar sources, is non-controllable and has high uncertainty and variability. We deal with non-Markovian dynamics through a Lagrangian relaxation, solving then a sequence of HJB PDEs associated with the system to find time-continuous optimal control and cost function. We introduce a monotone scheme to avoid spurious oscillations. Finally, we study the usefulness of extra system storage capacity.
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