Although the full form of the Rayleigh–Plesset (RP) equation more accurately depicts the bubble behavior in a cavitating flow than its reduced form, it finds much less application than the latter in the computational fluid dynamic (CFD) simulation due to its high stiffness. The traditional variable time-step scheme for the full form RP equation is difficult to be integrated with the CFD program since it requires a tiny time step at the singularity point for convergence and this step size may be incompatible with time marching of conservation equations. This paper presents two stable and efficient numerical solution schemes based on the finite difference method and Euler method so that the full-form RP equation can be better accepted by the CFD program. By employing a truncation bubble radius to approximate the minimum bubble size in the collapse stage, the proposed schemes solve for the bubble radius and wall velocity in an explicit way. The proposed solution schemes are more robust for a wide range of ambient pressure profiles than the traditional schemes and avoid excessive refinement on the time step at the singularity point. Since the proposed solution scheme can calculate the effects of the second-order term, liquid viscosity, and surface tension on the bubble evolution, it provides a more accurate estimation of the wall velocity for the vaporization or condensation rate, which is widely used in the cavitation model in the CFD simulation. The legitimacy of the solution schemes is manifested by the agreement between the results from these schemes and established ones from the literature. The proposed solution schemes are more robust in face of a wide range of ambient pressure profiles.
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, Structural Dynamic Problems in Time Domain under Uncertainty - An Interval Finite Element Approach
An analysis of the structural dynamic response under uncertainty is
presented. Uncertainties in load and material are modelled as intervals
exploiting the interval finite element method (IFEM). To reduce overestimation
and increase the computational efficiency of the solution, we do not solve the
dynamic problem by an explicit step-by-step time integration scheme. Instead,
our approach solves for the structural variables in the whole time domain
simultaneously by an implicit scheme using discrete Fourier transform and its
inverse (DFT and IDFT). Non-trivial initial conditions are handled by
modifying the right-hand side of the governing equation. To further reduce
overestimation, a new decomposition strategy is applied to the IFEM matrices,
and both primary and derived quantities are solved simultaneously. The final
solution is obtained using an iterative enclosure method, and in our numerical
examples the exact solution is enclosed at minimal computational cost.
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- Award ID(s):
- 1634483
- NSF-PAR ID:
- 10181603
- Date Published:
- Journal Name:
- International journal of reliability and safety
- Volume:
- 12
- Issue:
- 1/2
- ISSN:
- 1479-3903
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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The Cable equation is one of the most fundamental equations for modeling neuronal dynamics. In this article, we consider a high order compact finite difference numerical solution for the fractional Cable equation, which is a generalization of the classical Cable equation by taking into account the anomalous diffusion in the movement of the ions in neuronal system. The resulting finite difference scheme is unconditionally stable and converges with the convergence order of
in maximum norm, 1‐norm and 2‐norm. Furthermore, we present a fast solution technique to accelerate Toeplitz matrix‐vector multiplications arising from finite difference discretization. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from required by traditional methods to without using any lossy compression, where and τ is the size of time step,and h is the size of space step. Moreover, we give a compact finite difference scheme and consider its stability analysis for two‐dimensional fractional Cable equation. The applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis. -
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Abstract In this article, we present a decentralized control scheme for regulating input‐affine nonlinear interconnected systems. In particular, we propose a codesign strategy to synthesize a control policy and an event‐triggering threshold at each subsystem of an interconnected system to simultaneously optimize the subsystem performance and reduce the computational burden on the controllers by enforcing aperiodic dynamic feedback. To this end, we formulate a differential game at every subsystem to design a decentralized control scheme in which we treat the control policy as the minimizing player and model the effect of interconnection inputs and the error introduced due to aperiodic feedback as a team of adversarial players. We then employ the solution to the proposed game for designing both the control policy and the event‐triggering threshold at each subsystem. With the proposed approach, we also derive the conditions that guarantee the input‐to‐state stability of the overall system by leveraging the well‐known small‐gain theorem. Moreover, we show that these conditions, expressed in terms of the attenuation constants and penalty matrices introduced in the formulated game, are obtained as linear inequalities even when the dynamics of the subsystems are nonlinear. Finally, we illustrate the applicability of the proposed scheme to regulate interconnected systems using numerical examples.
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Monte Carlo (MC) methods are important computational tools for molecular structure optimizations and predictions. When solvent effects are explicitly considered, MC methods become very expensive due to the large degree of freedom associated with the water molecules and mobile ions. Alternatively implicit-solvent MC can largely reduce the computational cost by applying a mean field approximation to solvent effects and meanwhile maintains the atomic detail of the target molecule. The two most popular implicit-solvent models are the Poisson-Boltzmann (PB) model and the Generalized Born (GB) model in a way such that the GB model is an approximation to the PB model but is much faster in simulation time. In this work, we develop a machine learning-based implicit-solvent Monte Carlo (MLIMC) method by combining the advantages of both implicit solvent models in accuracy and efficiency. Specifically, the MLIMC method uses a fast and accurate PB-based machine learning (PBML) scheme to compute the electrostatic solvation free energy at each step. We validate our MLIMC method by using a benzene-water system and a protein-water system. We show that the proposed MLIMC method has great advantages in speed and accuracy for molecular structure optimization and prediction.more » « less
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Summary We present a general framework to solve elastodynamic problems by means of the virtual element method (VEM) with explicit time integration. In particular, the VEM is extended to analyze nearly incompressible solids using the B‐bar method. We show that, to establish a B‐bar formulation in the VEM setting, one simply needs to modify the stability term to stabilize only the deviatoric part of the stiffness matrix, which requires no additional computational effort. Convergence of the numerical solution is addressed in relation to stability, mass lumping scheme, element size, and distortion of arbitrary elements, either convex or nonconvex. For the estimation of the critical time step, two approaches are presented, ie, the maximum eigenvalue of a system of mass and stiffness matrices and an effective element length. Computational results demonstrate that small edges on convex polygonal elements do not significantly affect the critical time step, whereas convergence of the VEM solution is observed regardless of the stability term and the element shape in both two and three dimensions. This extensive investigation provides numerical recipes for elastodynamic VEMs with explicit time integration and related problems.
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1504/IJRS.2018.092516
