We consider a multirate iterative scheme for the quasi-static Biot equations modelling the coupled flow and geomechanics in a porous medium. The iterative scheme is based on undrained splitting where the flow and mech...
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ISBN:
(纸本)9783319399294;9783319399270
We consider a multirate iterative scheme for the quasi-static Biot equations modelling the coupled flow and geomechanics in a porous medium. The iterative scheme is based on undrained splitting where the flow and mechanics equations are decoupled with the mechanics solve followed by the pressure solve. The multirate scheme proposed here uses different time steps for the two equations, that is, uses q flow steps for each coarse mechanics step and may be interpreted as using a regularization parameter for the mechanics equation. We prove the convergence of the scheme and the proof reveals the appropriate regularization parameter and also the effect of the number of flow steps within coarse mechanics step on the convergence rate.
We present the numerical solution of a hydrodynamics model of flocking using a suitable modified semi-implicit discontinuous Galerkin method. The investigated model describing the dynamics of flocks of birds or other ...
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ISBN:
(纸本)9783319399294;9783319399270
We present the numerical solution of a hydrodynamics model of flocking using a suitable modified semi-implicit discontinuous Galerkin method. The investigated model describing the dynamics of flocks of birds or other individual entities forming herds or swarms was introduced by Fornasier et al. (Physica D 240(1): 2131, 2011). The main idea of this model comes from the well known Cucker-Smale model. The resulting equations consist of the Euler equations for compressible flow with an additional non-local non-linear source term. The model is discretized by the semi-implicit discontinuous Galerkin method for the compressible Euler equations of Feistauer and Kucera (J Comput Phys 224(1): 208-221, 2007). We show that with a suitable treatment of the source term we can use this method for models like the model of flocking and find a numerical solution very efficiently.
We apply continuous and discontinuous Galerkin time discretization together with standard finite element method for space discretization to the heat equation. For the numerical solution arising from these discretizati...
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ISBN:
(纸本)9783319399294;9783319399270
We apply continuous and discontinuous Galerkin time discretization together with standard finite element method for space discretization to the heat equation. For the numerical solution arising from these discretizations we present a guaranteed and fully computable a posteriori error upper bound. Moreover, we present local asymptotic efficiency estimate of this bound.
A variant of a nonlinear FETI-DP domain decomposition method is considered. It is combined with a parallel algebraic multigrid method (BoomerAMG) in a way which completely removes sparse direct solvers from the algori...
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ISBN:
(纸本)9783319399294;9783319399270
A variant of a nonlinear FETI-DP domain decomposition method is considered. It is combined with a parallel algebraic multigrid method (BoomerAMG) in a way which completely removes sparse direct solvers from the algorithm. Scalability to 524,288 MPI ranks is shown for linear elasticity and nonlinear hyperelasticity using more than half of the JUQUEEN supercomputer (JSC, Julich;TOP500 rank: 11th).
There is wide agreement that the accuracy of turbulence models suffer from their sensitivity with respect to physical input data, the uncertainties of user-elected parameters, as well as the model inadequacy. However,...
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ISBN:
(数字)9783319282626
ISBN:
(纸本)9783319282626;9783319282602
There is wide agreement that the accuracy of turbulence models suffer from their sensitivity with respect to physical input data, the uncertainties of user-elected parameters, as well as the model inadequacy. However, the application of Bayesian inference to systematically quantify the uncertainties in parameters, by means of exploring posterior probability density functions (PPDFs), has been hindered by the prohibitively daunting computational cost associated with the large number of model executions, in addition to daunting computation time per one turbulence simulation. In this effort, we perform in this paper an adaptive hierarchical sparse grid surrogate modeling approach to Bayesian inference of large eddy simulation (LES). First, an adaptive hierarchical sparse grid surrogate for the output of forward models is constructed using a relatively small number of model executions. Using such surrogate, the likelihood function can be rapidly evaluated at any point in the parameter space without simulating the computationally expensive LES model. This method is essentially similar to those developed in Zhang et al. (Water Resour Res 49: 6871-6892, 2013) for geophysical and groundwater models, but is adjusted and applied here for a much more challenging problem of uncertainty quantification of turbulence models. Through a numerical demonstration of the Smagorinsky model of two-dimensional flow around a cylinder at sub-critical Reynolds number, our approach is proven to significantly reduce the number of costly LES executions without losing much accuracy in the posterior probability estimation. Here, the model parameters are calibrated against synthetic data related to the mean flow velocity and Reynolds stresses at different locations in the flow wake. The influence of the user-elected LES parameters on the quality of output data will be discussed.
The Reduced Basis Method (RBM) (Rozza et al., Archiv Comput Methods Eng 15: 229-275, 2008) generates low-order models for efficient evaluation of parametrized PDEs in many-query and real-time contexts. It can be seen ...
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ISBN:
(纸本)9783319399294;9783319399270
The Reduced Basis Method (RBM) (Rozza et al., Archiv Comput Methods Eng 15: 229-275, 2008) generates low-order models for efficient evaluation of parametrized PDEs in many-query and real-time contexts. It can be seen as a parametric model reduction method (Benner et al., SIAM Rev 57(4): 483-531, 2015), where greedy selection is combined with a projection space composed of solution snapshots. The approximation quality is certified by using rigorous error estimators. We apply the RBM to systems of Maxwell's equations arising from electrical circuits. Using microstrip models, the input-output behaviour of the interconnect structures is approximated for a certain frequency range. Typically, an output is given by a linear functional, but in the case of impedance parameters (also called Z-parameters), the output is quadratic. An expanded formulation is used to rewrite the system to compliant form, i.e., a form, where the input and output are identical. This enables fast convergence in the approximation error and thus very low reduced model sizes. A numerical example from the microwave regime shows the advantage of this approach.
In the present paper we detail the implementation of the Virtual Element Method for two dimensional elliptic equations in primal and mixed form with variable coefficients.
ISBN:
(纸本)9783319416403;9783319416380
In the present paper we detail the implementation of the Virtual Element Method for two dimensional elliptic equations in primal and mixed form with variable coefficients.
Fractional flow reserve (FFR) is the golden standard for making decision on surgical treatment of coronary vessels with multiple stenosis. Clinical measurements of FFR require expensive invasive procedure with endovas...
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ISBN:
(纸本)9783319399294;9783319399270
Fractional flow reserve (FFR) is the golden standard for making decision on surgical treatment of coronary vessels with multiple stenosis. Clinical measurements of FFR require expensive invasive procedure with endovascular ultrasound probe. In this work a method of FFR simulation is considered. It is based on modelling 1D haemodynamics in patient-specific coronary vessels network reconstructed from CT scans. In contrast to our previous studies we used explicit minimum oscillating 2nd order characteristic method for internal nodes and 2nd order approximation of compatibility conditions for discretization of boundary conditions in junctions. The model is applied for simulating the change of FFR due to variability of the vessels elasticity and autoregulation response rate.
This work is devoted to the Directional Do-Nothing (DDN) condition as an outflow boundary condition for the incompressible Navier-Stokes equation. In contrast to the Classical Do-Nothing (CDN) condition, we have stabi...
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ISBN:
(纸本)9783319399294;9783319399270
This work is devoted to the Directional Do-Nothing (DDN) condition as an outflow boundary condition for the incompressible Navier-Stokes equation. In contrast to the Classical Do-Nothing (CDN) condition, we have stability, existence of weak solutions and, in the case of small data, also uniqueness. We derive an a priori error estimate for this outflow condition for finite element discretizations with inf-sup stable pairs. Stabilization terms account for dominant convection and the divergence free constraint. Numerical examples demonstrate the stability of the method.
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