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Sensitivity analysis is routinely performed on simplified surrogate models as the cost of such analysis on the original model may be prohibitive. Little is known in general about the induced bias on the sensitivity results. Within the framework of chemical kinetics, we provide a full justification of the above approach in the case of variance based methods provided the surrogate model results from the original one through the thermodynamic limit. We also provide illustrative numerical examples in context of a Michaelis--Menten system and a biochemical reaction network describing a genetic oscillator., 19 pages

Granular flows occur in a wide range of situations of practical interest to industry, in our natural environment and in our everyday lives. This paper focuses on granular flow in the so-called inertial regime, when the rheology is independent of the very large particle stiffness. Such flows have been modelled with the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology, which postulates that the bulk friction coefficient $\unicode[STIX]{x1D707}$ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction $\unicode[STIX]{x1D719}$ are functions of the inertial number $I$ only. Although the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology has been validated in steady state against both experiments and discrete particle simulations in several different geometries, it has recently been shown that this theory is mathematically ill-posed in time-dependent problems. As a direct result, computations using this rheology may blow up exponentially, with a growth rate that tends to infinity as the discretization length tends to zero, as explicitly demonstrated in this paper for the first time. Such catastrophic instability due to ill-posedness is a common issue when developing new mathematical models and implies that either some important physics is missing or the model has not been properly formulated. In this paper an alternative to the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology that does not suffer from such defects is proposed. In the framework of compressible $I$-dependent rheology (CIDR), new constitutive laws for the inertial regime are introduced; these match the well-established $\unicode[STIX]{x1D707}(I)$ and $\unicode[STIX]{x1D6F7}(I)$ relations in the steady-state limit and at the same time are well-posed for all deformations and all packing densities. Time-dependent numerical solutions of the resultant equations are performed to demonstrate that the new inertial CIDR model leads to numerical convergence towards physically realistic solutions that are supported by discrete element method simulations.

In this paper we analyze the convergence of the domain decomposition method applied to transport problems on networks. In particular, we derive estimates for the number of required iterations for linear problems. These estimates can be used to determine when the implementation of domain decomposition methods would be beneficial for this type of problems.

One-dimensional (1D) modeling is a widely adopted approach for studying wave propagation phenomena in the arterial system. Despite the frequent use of the Windkessel (WK) model to prescribe outflow boundary conditions for 1D arterial tree models, it remains unclear to what extent the inherent limitation of the WK model in describing wave propagation in distal vasculatures affect hemodynamic variables simulated at the arterial level. In the present study, a 1D model of the arterial tree was coupled respectively with a WK boundary model and a structured-tree (ST) boundary model, yielding two types of arterial tree models. The effective resistances, compliances and inductances of the WK and ST boundary models were matched to facilitate quantitative comparisons. Obtained results showed that pressure/flow waves simulated by the two models were comparable in the aorta, whereas, their discrepancies increased towards the periphery. Wave analysis revealed that the differences in reflected waves generated by the boundary models were the major sources of pressure wave discrepancies observed in large arteries. Additional simulations performed under aging conditions demonstrated that arterial stiffening with age enlarged the discrepancies, but with the effects being partly counteracted by physiological aortic dilatation with age. These findings suggest that the method adopted for modeling the outflow boundary conditions has considerable influence on the performance of a 1D arterial tree model, with the extent of influence varying with the properties of the arterial system.

The complexity and size of state-of-the-art cell models have significantly increased in part due to the requirement that these models possess complex cellular functions which are thought-but not necessarily proven-to be important. Modern cell models often involve hundreds of parameters; the values of these parameters come, more often than not, from animal experiments whose relationship to the human physiology is weak with very little information on the errors in these measurements. The concomitant uncertainties in parameter values result in uncertainties in the model outputs or quantities of interest (QoIs). Global sensitivity analysis (GSA) aims at apportioning to individual parameters (or sets of parameters) their relative contribution to output uncertainty thereby introducing a measure of influence or importance of said parameters. New GSA approaches are required to deal with increased model size and complexity; a three-stage methodology consisting of screening (dimension reduction), surrogate modeling, and computing Sobol' indices, is presented. The methodology is used to analyze a physiologically validated numerical model of neurovascular coupling which possess 160 uncertain parameters. The sensitivity analysis investigates three quantities of interest, the average value of [Formula: see text] in the extracellular space, the average volumetric flow rate through the perfusing vessel, and the minimum value of the actin/myosin complex in the smooth muscle cell. GSA provides a measure of the influence of each parameter, for each of the three QoIs, giving insight into areas of possible physiological dysfunction and areas of further investigation.

This study develops non-pulsatile and pulsatile models for the prediction of blood flow and pressure during head-up tilt. This test is used to diagnose potential pathologies within the autonomic control system, which acts to keep the cardiovascular system at homeostasis. We show that mathematical modeling can be used to predict changes in cardiac contractility, vascular resistance, and arterial compliance, quantities that cannot be measured but are useful to assess the system’s state. These quantities are predicted as time-varying parameters modeled using piecewise linear splines. Having models with various levels of complexity formulated with a common set of parameters, allows us to combine long-term non-pulsatile simulations with pulsatile simulations on a shorter time-scale. We illustrate results for a representative subject tilted head-up from a supine position to a $$60^{\circ }$$ angle. The tilt is maintained for 5 min before the subject is tilted back down. Results show that if volume data is available for all vascular compartments three parameters can be identified, cardiovascular resistance, vascular compliance, and ventricular contractility, whereas if model predictions are made against arterial pressure and cardiac output data alone, only two parameters can be estimated either resistance and contractility or resistance and compliance.

Global sensitivity analysis (GSA) is used to quantify the influence of uncertain variables in a mathematical model. Prior to performing GSA, the user must specify (or implicitly assume), a probability distribution to model the uncertainty, and possibly statistical dependencies, of the variables. Determining this distribution is challenging in practice as the user has limited and imprecise knowledge of the uncertain variables. This article analyzes the robustness of the Sobol' indices, a commonly used tool in GSA, to changes in the distribution of the uncertain variables. A method for assessing such robustness is developed which requires minimal user specification and no additional evaluations of the model. Theoretical and computational aspects of the method are considered and illustrated through examples., Comment: 20 pages

The formation of singularities in relativistic flows is not well understood. Smooth solutions to the relativistic Euler equations are known to have a finite lifespan; the possible breakdown mechanisms are shock formation, violation of the subluminal conditions and mass concentration. We propose a new hybrid Glimm/central-upwind scheme for relativistic flows. The scheme is used to numerically investigate, for a family of problems, which of the above mechanisms is involved.

SUMMARY We discuss the implementation and calibration of a new generalized structured tree boundary condition for hemodynamics. The main idea is to approximate the impedance corresponding to the vessels downstream from a specific outlet. Unlike previous impedance conditions, the one considered here is applicable to general transient flows as opposed to periodic ones only. The physiological character of the approach significantly simplifies calibration. We also describe a novel way to incorporate autoregulation mechanisms in structured arterial trees at minimal computational cost. The strength of the approach is illustrated and validated on several examples through comparison with clinical data. Copyright © 2014 John Wiley & Sons, Ltd.

The global sensitivity analysis of time-dependent processes requires history-aware approaches. We develop for that purpose a variance-based method that leverages the correlation structure of the problems under study and employs surrogate models to accelerate the computations. The errors resulting from fixing unimportant uncertain parameters to their nominal values are analyzed through a priori estimates. We illustrate our approach on a harmonic oscillator example and on a nonlinear dynamic cholera model., Comment: 28 Pages; revised version; accepted for publication in Reliability Engineering & System Safety

It is commonly assumed that perfusion in a given cerebral territory can be inferred from Blood Flow Velocity (BFV) measurements in the corresponding stem artery. In order to test this hypothesis, we construct a cerebral blood flow (CBF) estimator based on transcranial Doppler (TCD) blood flow velocity and ten other easily available patient characteristics and clinical parameters. A total of 261 measurements were collected from 88 older patients. The estimator is based on local regression (Random Forest). Its performance is analyzed against baseline CBF from 3-D pseudocontinuous arterial spin labeling (pCASL) magnetic resonance imaging (MRI). Patient specific CBF predictions are of poor quality (r = 0.41 and p-value = 4.5 × 10−12); the hypothesis is thus not clearly supported by evidence.

The equations of planar motion of a mass attached to two anchored massless springs form a symmetric Hamiltonian system. The system has a single dimensionless parameter L, corresponding to the spacing between the anchors. For L > 1, there is a stable equilibrium at which the springs are in tension and lie on a line, but for L < 1, this equilibrium has both springs in compression and is unstable. However, there are then two stable equilibria at which both springs carry no force. Oscillations are studied in both regimes, but more systematically in the tension case, where techniques of bifurcation theory, numerical approximation and numerical simulation are used to explore the rich variety of periodic solutions.

The computation of moving curves by the level set method typically requires reinitializations of the underlying level set function. Two types of reinitialization methods are studied: a high order ''PDE'' approach and a second order Fast Marching method. Issues related to the efficiency and implementation of both types of methods are discussed, with emphasis on the tube/narrow band implementation and accuracy considerations. The methods are also tested and compared. Fast Marching reinitialization schemes are faster but limited to second order, PDE based reinitialization schemes can easily be made more accurate but are slower, even with a tube/narrow band implementation.

Friction plays a fundamental role in the mechanics of granular materials. Two problems are considered: (i) heap formation and (ii) granular flow. Both problems admit closely related mathematical models. In each case, analytical and numerical difficulties are discussed. Efficient and reliable numerical methods are proposed and implemented. The results are illustrated by several computational experiments.

Similarity solutions are constructed for the flow of granular materials through hoppers. Unlike previous work, the present approach applies to nonaxisymmetric containers. The model involves ten unknowns (stresses, velocity, and plasticity function) determined by nine nonlinear first order partial differential equations together with a quadratic algebraic constraint (yield condition). A pseudospectral discretization is applied; the resulting problem is solved with a trust region method. The important role of the hopper geometry on the flow is illustrated by several numerical experiments of industrial relevance.

A simple model for the determination of the shape of large granular piles in complicated geometries is discussed. An eikonal formulation of the problem is proposed. Two distinct cases arise. In cylindrical geometries, i.e., if both container and possible obstacles have vertical walls, the problem is equivalent to a two-dimensional travel time problem with obstacles, while in general geometries, this analogy breaks down. In the first case, classical one-sided discretizations are generalized to handle obstacles without loss in accuracy. In the second case, a fast and efficient numerical method is proposed, implemented and tested. The discrete problems are solved through fast marching.

When a fine powder is dumped into a silo, the gas trapped by the particles will slowly escape by diffusing through the material. The corresponding uneven gas pressure distribution creates a body force that is taken into account through Darcy's law. By using spatial averaging, the formulation, even though essentially one-dimensional in space, includes effects due the geometry of the container. An efficient and robust numerical scheme based on a differential algebraic equation formulation is proposed and implemented. Various computational results are presented and discussed to establish the validity of the approach.

The problem of determining the steady state flow of granular materials in silos under the action of gravity is considered. In the case of Mohr–Coulomb materials, the stress equations correspond to a system of hyperbolic conservation laws with source terms and boundary conditions. A higher order discontinuous Galerkin method is proposed and implemented for the numerical resolution of those equations. The efficiency of the approach is illustrated by the computation of the stress fields induced in silos with sharp changes of the wall angle.

The flow of granular materials in hoppers is studied. In industrial applications, inserts of various sizes and shapes are often used to improve the flow properties and get rid of undesirable effects such as material sticking to the walls, funnel flow, arching, etc. We study the case of inverted conical inserts in conical hoppers. In spite of the complexity of the phenomenon as observed in practice, existing methods assume in general a radial structure of the stress and velocity fields. A new numerical approach to the problem of designing “optimal” inserts is proposed and tested. It allows for non-purely radial solutions. General comments about the overall approach and its relationship with experiments are offered.

This paper is the third of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed. In this paper, we consider multidimensional flux-splitting monotone schemes defined on non-Cartesian grids. We identify those schemes which are consistent and prove that the $L^\infty(0,T;L^1(\Bbb{R}^d))$-norm of the error goes to zero as $(\Delta x)^{1/2}$ when the discretization parameter $\Delta x$ goes to zero. Moreover, we show that nonconsistent schemes can converge at optimal rates of $(\Delta x)^{1/2}$ because (i) the conservation form of the schemes and (ii) the so-called consistency of the numerical fluxes allow the regularity properties of the approximate solution to compensate for their lack of consistency.

We present numerical computations for the flow of granular materials in hoppers. The algorithm is based on a cellular automaton approach and takes into account various microscopic effects, including “orientational effects”. The formation and the dynamics of density waves, recently discovered in physical experiments, is observed and analyzed computationally.

In this paper, new a posteriors error estimates for the shock-capturing streamline diffusion (SCSD) method and the shock-capturing discontinuous galerkin (SCDG) method for scalar conservation laws are obtained. These estimates are then used to prove that the SCSD method and the SCDG method converge to the entropy solution with a rate of at least $h^{{1 / 8}} $ and $h^{{1 / 8}} $, respectively, in the $L^\infty (L^1 )$-norm. The triangulations are made of general acute simplices and the approximate solution is taken to be piecewise a polynomial of degree k. The result is independent of the dimension of the space.

The goal of the paper is to bring to the attention of the computational community a long overlooked, very simple, acceleration method that impressively speeds up explicit time-stepping schemes, at essentially no extra cost. The authors explain the basis of the method, namely stabilization via wisely chosen inner steps (stages), justify it for linear problems, and spell out how simple it is to incorporate in any explicit code for parabolic problems. Finally, we demonstrate its performance on the (linear) heat equation as well as on the (non-linear) classical Stefan problem, by comparing it with standard implicit schemes (employing SOR or Newton iterations). The results show that super-time-stepping is more efficient than the implicit schemes in that it runs at least as fast, it is of comparable or better accuracy, and it is, of course, much easier to program (and to parallelize for distributed computing).

In the Calculus of Variations, several notions of convexity have emerged, corresponding to different properties of the functionals to be minimized. The relations between these various notions are not yet fully understood. In this context, we present a numerical study of quasiconvexity for some functions of the type f(ξ) = g(|ξ|2, det ξ), where Ξ is a 2×2-matrix. The corresponding global optimization problems are solved using a simulated annealing-like algorithm. The computations strongly indicate that the considered functions are quasiconvex if and only if they are rank-one convex. The relation to Morrey's conjecture, various applications and implementation problems are discussed.

The description of equilibria of shape memory alloys or other ordered materials gives rise to nonconvex variational problems. In this paper, a two-dimensional model of such materials is studied. Due to the fact that the corresponding functional has two symmetry-related (martensitic) energy wells, the numerical approximation of the deformation gradient does not converge, but tends to oscillate between the two wells, as the size of the mesh is refined. These oscillations may be interpreted in terms of microstructures. Using a nonconforming $\mathcal{P}_1 $ finite element, an estimate is given for the rate of convergence of the probability for the approximated deformation to have its gradient “near” one of the two (martensitic) wells.