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A boundary integral method for modelling vibroacoustic energy distributions in uncertain built up structures
A phase-space boundary integral method is developed for modelling stochastic high-frequency acoustic and vibrational energy transport in both single and multi-domain problems. The numerical implementation is carried out using the collocation method in both the position and momentum phase-space variables. One of the major developments of this work is the systematic convergence study, which demonstrates that the proposed numerical schemes exhibit convergence rates that could be expected from theoretical estimates under the right conditions. For the discretisation with respect to the momentum variable, we employ spectrally convergent basis approximations using both Legendre polynomials and Gaussian radial basis functions. The former have the advantage of being simpler to apply in general without the need for preconditioning techniques. The Gaussian basis is introduced with the aim of achieving more efficient computations in the weak noise case with near-deterministic dynamics. Numerical results for a series of coupled domain problems are presented, and demonstrate the potential for future applications to larger scale problems from industry
On the stability of projection methods for the incompressible Navier-Stokes equations based on high-order discontinuous Galerkin discretizations
The present paper deals with the numerical solution of the incompressible
Navier-Stokes equations using high-order discontinuous Galerkin (DG) methods
for discretization in space. For DG methods applied to the dual splitting
projection method, instabilities have recently been reported that occur for
coarse spatial resolutions and small time step sizes. By means of numerical
investigation we give evidence that these instabilities are related to the
discontinuous Galerkin formulation of the velocity divergence term and the
pressure gradient term that couple velocity and pressure. Integration by parts
of these terms with a suitable definition of boundary conditions is required in
order to obtain a stable and robust method. Since the intermediate velocity
field does not fulfill the boundary conditions prescribed for the velocity, a
consistent boundary condition is derived from the convective step of the dual
splitting scheme to ensure high-order accuracy with respect to the temporal
discretization. This new formulation is stable in the limit of small time steps
for both equal-order and mixed-order polynomial approximations. Although the
dual splitting scheme itself includes inf-sup stabilizing contributions, we
demonstrate that spurious pressure oscillations appear for equal-order
polynomials and small time steps highlighting the necessity to consider inf-sup
stability explicitly.Comment: 31 page
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