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Register a new userSurface Impedance Determination via Numerical Resolution of the Inverse Helmholtz Problem
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Assigning boundary conditions, such as acoustic impedance, to the frequency domain thermoviscous wave equations (TWE), derived from the linearized Navier-Stokes equations (LNSE) poses a Helmholtz problem, solution to which yields a discrete set of complex eigenfunctions and eigenvalue pairs. The proposed method -- the inverse Helmholtz solver (iHS) -- reverses such procedure by returning the value of acoustic impedance at one or more unknown impedance boundaries (IBs) of a given domain, via spatial integration of the TWE for a given real-valued frequency with assigned conditions on other boundaries. The iHS procedure is applied to a second-order spatial discretization of the TWEs on an unstructured staggered grid arrangement. Only the momentum equation is extended to the center of each IB face where pressure and velocity components are co-located and treated as unknowns. The iHS is finally closed via assignment of the surface gradient of pressure phase over the IBs, corresponding to assigning the shape of the acoustic waveform at the IB. The iHS procedure can be carried out independently for different frequencies, making it embarrassingly parallel, and able to return the complete broadband complex impedance distribution at the IBs in any desired frequency range to arbitrary numerical precision. The iHS approach is first validated against Rotts theory for viscous rectangular and circular ducts. The impedance of a toy porous cavity with a complex geometry is then reconstructed and validated with companion fully compressible unstructured Navier-Stokes simulations resolving the cavity geometry. Verification against one-dimensional impedance test tube calculations based on time-domain impedance boundary conditions (TDIBC) is also carried out. Finally, results from a preliminary analysis of a thermoacoustically unstable cavity are presented.
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Assigning homogeneous boundary conditions, such as acoustic impedance, to the thermoviscous wave equations (TWE) derived by transforming the linearized Navier-Stokes equations (LNSE) to the frequency domain yields a so-called Helmholtz solver, whose output is a discrete set of complex eigenfunction and eigenvalue pairs. The proposed method -- the inverse Helmholtz solver (iHS) -- reverses such procedure by returning the value of acoustic impedance at one or more unknown impedance boundaries (IBs) of a given domain via spatial integration of the TWE for a given real-valued frequency with assigned conditions on other boundaries. The iHS procedure is applied to a second-order spatial discretization of the TWEs derived on an unstructured grid with staggered grid arrangement. The momentum equation only is extended to the center of each IB face where pressure and velocity components are co-located and treated as unknowns. One closure condition considered for the iHS is the assignment of the surface gradient of pressure phase over the IBs, corresponding to assigning the shape of the acoustic waveform at the IB. The iHS procedure is carried out independently for each frequency in order to return the complete broadband complex impedance distribution at the IBs in any desired frequency range. The iHS approach is first validated against Rotts theory for both inviscid and viscous, rectangular and circular ducts. The impedance of a geometrically complex toy cavity is then reconstructed and verified against companion full compressible unstructured Navier-Stokes simulations resolving the cavity geometry and one-dimensional impedance test tube calculations based on time-domain impedance boundary conditions (TDIBC). The iHS methodology is also shown to capture thermoacoustic effects, with reconstructed impedance values quantitatively in agreement with thermoacoustic growth rates.
We consider a size-structured model for cell division and address the question of determining the division (birth) rate from the measured stable size distribution of the population. We propose a new regularization technique based on a filtering approach. We prove convergence of the algorithm and validate the theoretical results by implementing numerical simulations, based on classical techniques. We compare the results for direct and inverse problems, for the filtering method and for the quasi-reversibility method proposed in [Perthame-Zubelli].
The infamous numerical sign problem poses a fundamental obstacle to long-time stochastic Wigner simulations in high dimensional phase space. Although the existing particle annihilation via uniform mesh (PAUM) significantly alleviates the sign problem when dimensionality D $le$ 4, the setting of regular grids gives rise to another challenge in data storage when D $ge$ 6 due to the curse of dimensionality. In this paper, we propose to use a recently developed adaptive particle annihilation, termed sequential-clustering particle annihilation via discrepancy estimation (SPADE), to overcome the numerical sign problem. SPADE consists of adaptive clustering of particles via controlling their number-theoretic discrepancies and independent random matching in each group, and may learn the minimal amount of particles that can accurately capture the oscillating nature of the Wigner function. Combining SPADE with a recently proposed variance reduction technique via the stationary phase approximation, we make the first attempt to simulate the transitions of hydrogen energy levels in 6-D phase space, where the feasibility of PAUM with sample sizes about $10^9$-$10^{10}$ has also been explored as a comparison.
This paper presents a topology optimization approach for surface flows, which can represent the viscous and incompressible fluidic motions at the solid/liquid and liquid/vapor interfaces. The fluidic motions on such material interfaces can be described by the surface Navier-Stokes equations defined on 2-manifolds or two-dimensional manifolds, where the elementary tangential calculus is implemented in terms of exterior differential operators expressed in a Cartesian system. Based on the topology optimization model for fluidic flows with porous medium filling the design domain, an artificial Darcy friction is added to the area force term of the surface Navier-Stokes equations and the physical area forces are penalized to eliminate their existence in the fluidic regions and to avoid the invalidity of the porous medium model. Topology optimization for steady and unsteady surface flows can be implemented by iteratively evolving the impermeability of the porous medium on the 2-manifolds, where the impermeability is interpolated by the material density derived from a design variable. The related partial differential equations are solved by using the surface finite element method. Numerical examples have been provided to demonstrate this topology optimization approach for surface flows, including the boundary velocity driven flows, area force driven flows and convection-diffusion flows.
The problem of the fictitious frequency spectrum resulting from numerical implementations of the boundary element method for the exterior Helmholtz problem is revisited. When the ordinary 3D free space Greens function is replaced by a modified Greens function, it is shown that these fictitious frequencies do not necessarily have to correspond to the internal resonance frequency of the object. Together with a recently developed fully desingularized boundary element method that confers superior numerical accuracy, a simple and practical way is proposed for detecting and avoiding these fictitious solutions. The concepts are illustrated with examples of a scattering wave on a rigid sphere.
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