Space-dependent Term Research Articles

Applicability of adiabatic approximation on neutron noise analysis in molten salt reactor

In this paper, the applicability of the adiabatic approximation on the neutron noise analysis in the molten salt reactor (MSR) is investigated. The neutronic model considering the fuel recirculation is established based on one-group neutron diffusion theory. In linear perturbation theory, the fluctuation equations are derived due to the smallness of the perturbation. Following the standard procedure, the kinetic equations for the fluctuations of amplitude factors, which are dependent on the unknown fluctuations of shape functions, are obtained. If the fluctuations of shape functions are neglected, the solutions of both the total neutron noise and its point kinetic term are inaccurate. In order to estimate the effects of the fluctuations of shape functions and the interference between the point kinetic and space-dependent terms, the adiabatic approximation is applied. The solutions are compared to the exact ones and those obtained in the point kinetic approximation to figure out the applicability of the adiabatic approximation. The results show that the adiabatic approximation is accurate enough for low velocities. With higher velocities, the solutions obtained in both the point kinetic and adiabatic approximations gradually deviate from the exact ones, whereas the adiabatic approximation can still provide good estimation of the frequency- and space-dependent behavior of neutron noise in the plateau region. In particular, the spatial oscillation of the amplitude of neutron noise can be well reconstructed. Moreover, the characteristic peak, which is related to the inverse of fuel recirculation, is absent in the point kinetic approximation but can be observed in the adiabatic approximation. Therefore, the adiabatic approximation can be regarded as an effective tool for reconstructing and interpreting the neutron noise in MSR.

Multiple surface reactions in arrays with applications to optical biosensors.

We analyze surface-volume reactions in the context of optical biosensors with arrays of reacting zones. For arrays having zones with the same rate constants, we consider a two-dimensional reacting zone boundary definition and quantify ligand depletion with the effective Damköhler number. We use asymptotics to obtain ligand depletion results for the one-dimensional case, and also compute results for the circular reacting zone case. For arrays having zones with different rate constants, depletion effects cannot be expressed as the product of time-dependent and space-dependent terms, and we propose two effective rate constant equations for this case.

Large-scale Gauss-Newton inversion of transient controlled-source electromagnetic measurement data using the model reduction framework

Transient data controlled-source electromagnetic measurements are usually interpreted via extracting few frequencies and solving the corresponding inverse frequency-domain problem. Coarse frequency sampling may result in loss of information and affect the quality of interpretation; however, refined sampling increases computational cost. Fitting data directly in the time domain has similar drawbacks, i.e., its large computational cost, in particular, when the Gauss-Newton (GN) algorithm is used for the misfit minimization. That cost is mainly comprised of the multiple solutions of the forward problem and linear algebraic operations using the Jacobian matrix for calculating the GN step. For large-scale 2.5D and 3D problems with multiple sources and receivers, the corresponding cost grows enormously for inversion algorithms using conventional finite-difference time-domain (FDTD) algorithms. A fast 3D forward solver based on the rational Krylov subspace (RKS) reduction algorithm using an optimal subspace selection was proposed earlier to partially mitigate this problem. We applied the same approach to reduce the size of the time-domain Jacobian matrix. The reduced-order model (ROM) is obtained by projecting a discretized large-scale Maxwell system onto an RKS with optimized poles. The RKS expansion replaces the time discretization for forward and inverse problems; however, for the same or better accuracy, its subspace dimension is much smaller than the number of time steps of the conventional FDTD. The crucial new development of this work is the space-time data compression of the ROM forward operator and decomposition of the ROM’s time-domain Jacobian matrix via chain rule, as a product of time- and space-dependent terms, thus effectively decoupling the discretizations in the time and parameter spaces. The developed technique can be equivalently applied to finely sampled frequency-domain data. We tested our approach using synthetic 2.5D examples of hydrocarbon reservoirs in the marine environment.

Reaction–Diffusion Systems and External Morphogen Gradients: The Two-Dimensional Case, with an Application to Skeletal Pattern Formation

We investigate a reaction-diffusion system consisting of an activator and an inhibitor in a two-dimensional domain. There is a morphogen gradient in the domain. The production of the activator depends on the concentration of the morphogen. Mathematically, this leads to reaction-diffusion equations with explicitly space-dependent terms. It is well known that in the absence of an external morphogen, the system can produce either spots or stripes via the Turing bifurcation. We derive first-order expansions for the possible patterns in the presence of an external morphogen and show how both stripes and spots are affected. This work generalizes previous one-dimensional results to two dimensions. Specifically, we consider the quasi-one-dimensional case of a thin rectangular domain and the case of a square domain. We apply the results to a model of skeletal pattern formation in vertebrate limbs. In the framework of reaction-diffusion models, our results suggest a simple explanation for some recent experimental findings in the mouse limb which are much harder to explain in positional-information-type models.

Space-time finite element approximation andnumerical solution of hereditary linearviscoelasticity problems

In this paper we suggest a fast numerical approach to treat problems of the hereditary linear viscoelasticity, which results in the system of elliptic partial differential equations in space variables, who's coefficients are Volterra integral operators of the second kind in time. We propose to approximate the relaxation kernels by the product of purely time- and space-dependent terms, which is achieved by their piecewise-polynomial space-interpolation. A priori error estimate was obtained and it was shown, that such approximation does not decrease the convergence order, when an interpolation polynomial is chosen of the same order as the shape functions for the spatial finite element approximation, while the computational effort is significantly reduced.

Stability of front solutions in inhomogeneous media

We use the FitzHugh–Nagumo (F–N) model to study front solutions in inhomogeneous media. Inhomogeneity is modeled by adding a space-dependent term, z( x), in the equations. Changes in z shift the cubic nullcline up and down in the phase-plane of the F–N model. This is used to mimic shifts in the cubic-shaped nullcline of a bidomain model of calcium waves in egg cells [Physica D, in press] caused by changes in the total calcium concentration. Conditions for the existence and stability of stationary front solutions for a general class of z functions are obtained analytically when the dynamics are piecewise linear. We show that spatial inhomogeneities cause pinning and oscillations of front solutions. This is best demonstrated when z( x) is a linear ramp. When the slope is large, a linear ramp breaks the translational invariance of the stationary front and stabilizes the front. The front becomes less stable as the slope decreases. At a critical slope, the front becomes unstable through a Hopf bifurcation beyond which oscillations in the front occur. This also occurs for nonlinear spatial inhomogeneities including a step increase with varying magnitude and steepness. This bifurcation has been found previously [Physica D 106 (1997) 270; Phys. Rev. E 55 (1997) 366] in a local analysis of the F–N model near a co-dimension two point (sometimes called a non-equilibrium Ising–Bloch (NIB) bifurcation) and with inhomogeneities of small magnitude. The analysis presented here applies globally in parameter space and is valid for inhomogeneities of arbitrary magnitude. We show that this bifurcation need not occur in conjunction with the NIB point. It is caused exclusively by spatial inhomogeneity. A rich variety of wave phenomena including front oscillation, front pinning, and front reflection are shown to occur in inhomogeneous media.

A General Approach to Stress Stiffening Effects on Flexible Multibody Dynamic Systems∗

Abstract This paper presents a general way of including stress stiffening effects in the dynamic simulation of flexible multibody systems that undergo large overall motion accompanied by small elastic deformation. Stress stiffening terms are derived from the internal virtual work that includes nonlinear terms of the strain-displacement relationship and reference stresses induced by existing loads before deformation. The resultant stress stiffness matrix for a flexible body is obtained through quasi-static structural analyses by applying D'Alembert inertia, joint reaction, and other applied loads. The DAIembert inertia loads arising from gross motion of the body reference frame are expressed as a combination of space-dependent terms and 12 sets of time-dependent terms. An iterative method is proposed to solve the resultant system equations of motion. Examples are presented to illustrate the method and procedures proposed.