504 research outputs found
Model Reduction for Multiscale Lithium-Ion Battery Simulation
In this contribution we are concerned with efficient model reduction for
multiscale problems arising in lithium-ion battery modeling with spatially
resolved porous electrodes. We present new results on the application of the
reduced basis method to the resulting instationary 3D battery model that
involves strong non-linearities due to Buttler-Volmer kinetics. Empirical
operator interpolation is used to efficiently deal with this issue.
Furthermore, we present the localized reduced basis multiscale method for
parabolic problems applied to a thermal model of batteries with resolved porous
electrodes. Numerical experiments are given that demonstrate the reduction
capabilities of the presented approaches for these real world applications
Mini-Workshop: Adaptive Methods for Control Problems Constrained by Time-Dependent PDEs
Optimization problems constrained by time-dependent PDEs (Partial Differential Equations) are challenging from a computational point of view: even in the simplest case, one needs to solve a system of PDEs coupled globally in time and space for the unknown solutions (the state, the costate and the control of the system). Typical and practically relevant examples are the control of nonlinear heat equations as they appear in laser hardening or the thermic control of flow problems (Boussinesq equations). Specifically for PDEs with a long time horizon, conventional time-stepping methods require an enormous storage of the respective other variables. In contrast, adaptive methods aim at distributing the available degrees of freedom in an a-posteriori-fashion to capture singularities and are, therefore, most promising
Comparison of POD reduced order strategies for the nonlinear 2D Shallow Water Equations
This paper introduces tensorial calculus techniques in the framework of
Proper Orthogonal Decomposition (POD) to reduce the computational complexity of
the reduced nonlinear terms. The resulting method, named tensorial POD, can be
applied to polynomial nonlinearities of any degree . Such nonlinear terms
have an on-line complexity of , where is the
dimension of POD basis, and therefore is independent of full space dimension.
However it is efficient only for quadratic nonlinear terms since for higher
nonlinearities standard POD proves to be less time consuming once the POD basis
dimension is increased. Numerical experiments are carried out with a two
dimensional shallow water equation (SWE) test problem to compare the
performance of tensorial POD, standard POD, and POD/Discrete Empirical
Interpolation Method (DEIM). Numerical results show that tensorial POD
decreases by times the computational cost of the on-line stage of
standard POD for configurations using more than model variables. The
tensorial POD SWE model was only slower than the POD/DEIM SWE model
but the implementation effort is considerably increased. Tensorial calculus was
again employed to construct a new algorithm allowing POD/DEIM shallow water
equation model to compute its off-line stage faster than the standard and
tensorial POD approaches.Comment: 23 pages, 8 figures, 5 table
Structure preserving reduced order modeling for gradient systems
Minimization of energy in gradient systems leads to formation of oscillatory
and Turing patterns in reaction-diffusion systems. These patterns should be
accurately computed using fine space and time meshes over long time horizons to
reach the spatially inhomogeneous steady state. In this paper, a reduced order
model (ROM) is developed which preserves the gradient dissipative structure.
The coupled system of reaction-diffusion equations are discretized in space by
the symmetric interior penalty discontinuous Galerkin (SIPG) method. The
resulting system of ordinary differential equations (ODEs) are integrated in
time by the average vector field (AVF) method, which preserves the energy
dissipation of the gradient systems. The ROMs are constructed by the proper
orthogonal decomposition (POD) with Galerkin projection. The nonlinear reaction
terms are computed efficiently by discrete empirical interpolation method
(DEIM). Preservation of the discrete energy of the FOMs and ROMs with POD-DEIM
ensures the long term stability of the steady state solutions. Numerical
simulations are performed for the gradient dissipative systems with two
specific equations; real Ginzburg-Landau equation and Swift-Hohenberg equation.
Numerical results demonstrate that the POD-DEIM reduced order solutions
preserve well the energy dissipation over time and at the steady state
Energy preserving model order reduction of the nonlinear Schr\"odinger equation
An energy preserving reduced order model is developed for two dimensional
nonlinear Schr\"odinger equation (NLSE) with plane wave solutions and with an
external potential. The NLSE is discretized in space by the symmetric interior
penalty discontinuous Galerkin (SIPG) method. The resulting system of
Hamiltonian ordinary differential equations are integrated in time by the
energy preserving average vector field (AVF) method. The mass and energy
preserving reduced order model (ROM) is constructed by proper orthogonal
decomposition (POD) Galerkin projection. The nonlinearities are computed for
the ROM efficiently by discrete empirical interpolation method (DEIM) and
dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and
mass are shown for the full order model (FOM) and for the ROM which ensures the
long term stability of the solutions. Numerical simulations illustrate the
preservation of the energy and mass in the reduced order model for the two
dimensional NLSE with and without the external potential. The POD-DMD makes a
remarkable improvement in computational speed-up over the POD-DEIM. Both
methods approximate accurately the FOM, whereas POD-DEIM is more accurate than
the POD-DMD
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