The Solvability of Mixed Value Problem for the First and Second Approximations of One-Dimensional Nonlinear System of Moment Equations with Microscopic Boundary Conditions

The paper gives a derivation of a new one-dimensional non-stationary nonlinear system of moment equations, that depend on the flight velocity and the surface temperature of an aircraft. Maxwell microscopic condition is approximated for the distribution function on moving boundary, when one fraction of molecules reflected from the surface specular and another fraction diffusely with Maxwell distribution. Moreover, macroscopic boundary conditions for the moment system of equations depend on evenness or oddness of approximation $${f}_{k}(t,x,c)$$ f k ( t , x , c ) , where $${f}_{k}(t,x,c)$$ f k ( t , x , c ) is partial expansion sum of the molecules distribution function over eigenfunctions of linearized collision operator around local Maxwell distribution. The formulation of initial and boundary value problem for the system of moment equations in the first and second approximations is described. Existence and uniqueness of the solution for the above-mentioned problem using macroscopic boundary conditions in the space of functions $$C\left(\left[0,T\right];{L}^{2}\left[-a,a\right]\right)$$ C 0 , T ; L 2 - a , a are proved.

Uncertainty propagation for deterministic models of biochemical networks using moment equations and the extended Kalman filter

Differential equation models of biochemical networks are frequently associated with a large degree of uncertainty in parameters and/or initial conditions. However, estimating the impact of this uncertainty on model predictions via Monte Carlo simulation is computationally demanding. A more efficient approach could be to track a system of low-order statistical moments of the state. Unfortunately, when the underlying model is nonlinear, the system of moment equations is infinite-dimensional and cannot be solved without a moment closure approximation which may introduce bias in the moment dynamics. Here, we present a new method to study the time evolution of the desired moments for nonlinear systems with polynomial rate laws. Our approach is based on solving a system of low-order moment equations by substituting the higher-order moments with Monte Carlo-based estimates from a small number of simulations, and using an extended Kalman filter to counteract Monte Carlo noise. Our algorithm provides more accurate and robust results compared to traditional Monte Carlo and moment closure techniques, and we expect that it will be widely useful for the quantification of uncertainty in biochemical model predictions.

Improvement and Assessment of the Plastic Collapse Bending Moment Equations in Circumferentially Cracked Pipe

The plastic collapse bending moment in a pipe cross-section with a circumferential crack is defined in ASME B&PV Code Section XI, Appendix C using simplified equilibrium equations by approximating the pipe mean radius Rm and the neutral axis angle β. In previous papers it was demonstrated by the authors that, for externally cracked pipes, those simplified equilibrium equations are not conservative and hence improved equations were developed and proposed which account for the cracked pipe ligament mean radius Rmc. In this paper, it is demonstrated that the accuracy of the collapse bending moment equation can be refined by taking into account the neutral axis position Yna of the cracked pipe section. This leads to exact collapse bending moment equations without any approximation on the pipe mean radius Rm nor on the neutral axis angle β. In this framework, it is shown that, for externally cracked pipes, the Appendix C equations could lead to more than 20% less conservative collapse bending moment than with the exact equations. An extended finite element method analysis completes this study to assess the relevance of the model used to determine the plastic collapse bending moment.

Data assimilation with multiple types of observation boreholes via the ensemble Kalman filter embedded within stochastic moment equations

We employ an approach based on the ensemble Kalman filter coupled with stochastic moment equations (MEs-EnKF) of groundwater flow to explore the dependence of conductivity estimates on the type of available information about hydraulic heads in a three-dimensional randomly heterogeneous field where convergent flow driven by a pumping well takes place. To this end, we consider three types of observation devices corresponding to (i) multi-node monitoring wells equipped with packers (Type A) and (ii) partially (Type B) and (iii) fully (Type C) screened wells. We ground our analysis on a variety of synthetic test cases associated with various configurations of these observation wells. Moment equations are approximated at second order (in terms of the standard deviation of the natural logarithm, Y, of conductivity) and are solved by an efficient transient numerical scheme proposed in this study. The use of an inflation factor imposed to the observation error covariance matrix is also analyzed to assess the extent at which this can strengthen the ability of the MEs-EnKF to yield appropriate conductivity estimates in the presence of a simplified modeling strategy where flux exchanges between monitoring wells and aquifer are neglected. Our results show that (i) the configuration associated with Type A monitoring wells leads to conductivity estimates with the (overall) best quality, (ii) conductivity estimates anchored on information from Type B and C wells are of similar quality, (iii) inflation of the measurement-error covariance matrix can improve conductivity estimates when a simplified flow model is adopted, and (iv) when compared with the standard Monte Carlo-based EnKF method, the MEs-EnKF can efficiently and accurately estimate conductivity and head fields.

Stationary Response of Nonlinear Vibration Energy Harvesters by Path Integration

A path integration procedure based on Gauss-Legendre integration scheme is developed to analyze probabilistic solution of nonlinear vibration energy harvesters (VEH) in this paper. First, traditional energy harvesters are briefly introduced and their non-dimensional governing and moment equations are given. These moment equations could be solved through the Runge-Kutta and Gaussian closure method. Then, the path integration method is expanded to three-dimensional situation, solving the probability density function (PDF) of VEH. Three illustrative examples are considered to evaluate the effectiveness of this method. The effectiveness of nonlinearity of traditional monostable VEH and a bistable VEH are further studied too. At the same time, Equivalent linearization method(EQL) and Monte Carlo simulation are employed too. The results indicate that three-dimensional path integration method can give satisfactory results for the global PDF, especially for the tail PDF, and they have better agreement with the simulation results than those of the EQL. In addition, the different degrees of hardening and softening behaviors of the PDFs occur when the nonlinearity coefficient increases and the bistable type is considered.