Continuous Sensitivity Equation Research Articles

Numerical approximation and uncertainty quantification for arterial blood flow models with viscoelasticity

The importance of the study of the blood flow equations is widely recognized as it is a tool to understand the circulatory system. Arteries and veins result to have both elastic and viscous behaviour. Models for the first case are much more studied as they result to be simpler and still satisfying if compared to experimental data. In this paper, we consider a model which encompasses both the elastic and viscoelastic response in arterial walls, respectively leading to a conservative and a non-conservative system. We present a second-order scheme based on the first-order Price-T scheme and the MUSCL-Hancock strategy. This approach automatically adapts to the above conservative and non-conservative cases.Then, we perform a Sensitivity Analysis (SA) based on the Continuous Sensitivity Equation Method (CSEM), whose aim is the study of how changes in the inputs of a model can affect its outputs. In particular, the sensitivity is defined as the derivative (with respect to an uncertain parameter a) of the solution of the system taken into consideration. Since the CSEM cannot be directly applied to discontinuous solutions, we add a source term to compensate the spikes associated to the Dirac delta functions that can arise in the sensitivity variables.One of the main applications of SA is uncertainty quantification, which is investigated for a Riemann problem as well as for a network of 37 arteries. Details on junctions for coupling two or more vessels are also given.

A modified sensitivity equation method for the Euler equations in presence of shocks

AbstractThe continuous sensitivity equation method allows to quantify how changes in the input of a partial differential equation (PDE) model affect the outputs, by solving additional PDEs obtained by differentiating the model. However, this method cannot be used directly in the framework of hyperbolic PDE systems with discontinuous solution, because it yields Dirac delta functions in the sensitivity solution at the location of state discontinuities. This difficulty is well known from theoretical viewpoint, but only a few works can be found in the literature regarding the possible numerical treatment. Therefore, we investigate in this study how classical numerical schemes for compressible Euler equations can be modified to account for shocks when computing the sensitivity solution. In particular, we propose the introduction of a source term, that allows to remove the spikes associated to the Dirac delta functions in the numerical solution. Numerical studies exhibit a strong impact of the numerical diffusion on the accuracy of this strategy. Therefore, we propose an anti‐diffusive numerical scheme coupled with the approximate Riemann solver of Roe for the state problem. For the sensitivity problem, two different numerical schemes are implemented and compared: one which takes into account the contact wave and another that neglects it. The effects of the numerical diffusion on the convergence of the schemes with respect to the grid are discussed. Finally, an application to uncertainty propagation is investigated and the different numerical schemes are compared.

A sensitivity study of relaxation parameter in Uzawa algorithm for the steady natural convection model

Purpose This paper aims to study the sensitivity of relaxation parameter in Uzawa method for the steady natural convection model. Design/methodology/approach Based on the continuous sensitivity equation method, associated sensitivity system is formed, which is solved by applying finite element method. Findings A decrease in sensitivity values of velocity, pressure and temperature is observed as the relaxation parameter increases. Originality/value The sensitivity study of relaxation parameter in Uzawa method is the first for the natural convection model, to the best of the authors’ knowledge. In fact, it is difficult to find an optimal relaxation parameter in the Uzawa method for the nonlinear problems. It is therefore important to understand how small changes in the relaxation parameter could affect the numerical solution of the nonlinear equations.

Continuum shape sensitivity analysis for aeroelastic gust using an arbitrary lagrangian-eulerian reference frame

Continuum Sensitivity Analysis (CSA), a method to determine response derivatives with respect to design variables, is derived here for the first time in an arbitrary Lagrangian-Eulerian (ALE) reference frame. CSA differentiates nonlinear governing system of equations to arrive at a linear system of partial differential continuum sensitivity equations (CSEs), here, for fluid-structure interaction (FSI). The CSEs and associated sensitivity boundary conditions are derived here for the first time for FSI, using the boundary velocity formulation, carefully distinguishing design velocity from flow velocity and ALE mesh velocity. Whereas boundary conditions must be differentiated using the material (total) derivative, it is sometimes advantageous to derive the CSEs using local (partial) derivatives. The benefit is that geometric sensitivity, known as design velocity, may not be required in the domain. It is shown here that this advantage is realized when the ALE frame undergoes only the rigid body motion associated with the structure to which it is attached. It is further shown that the advantage is not realized when the ALE mesh deforms due to the flexible motion of the fluid-structure interface. The equations for the transient gust response of a two-dimensional airfoil in compressible flow, flexibly attached to a rigid body mass, are presented as a model problem to illustrate a detailed derivation.

Sensitivity Analysis of Hydraulic Head to Locations of Model Boundaries

AbstractSensitivity analysis is an important component of many model activities in hydrology. Numerous studies have been conducted in calculating various sensitivities. Most of these sensitivity analysis focus on the sensitivity of state variables (e.g., hydraulic head) to parameters representing medium properties such as hydraulic conductivity or prescribed values such as constant head or flux at boundaries, while few studies address the sensitivity of the state variables to some shape parameters or design parameters that control the model domain. Instead, these shape parameters are typically assumed to be known in the model. In this study, based on the flow equation, we derive the equation (and its associated initial and boundary conditions) for sensitivity of hydraulic head to shape parameters using continuous sensitivity equation (CSE) approach. These sensitivity equations can be solved numerically in general or analytically in some simplified cases. The approach has been demonstrated through two examples and the results are compared favorably to those from analytical solutions or numerical finite difference methods with perturbed model domains, while numerical shortcomings of the finite difference method are avoided.

Sensitivity Analysis and Numerical Diffusion Effects for Hyperbolic PDE Systems with Discontinuous Solutions. The Case of Barotropic Euler Equations in Lagrangian Coordinates

Sensitivity analysis (SA) is the study of how the output of a mathematical model is affected by changes in the inputs. SA is widely studied, due to its many applications: uncertainty quantifi-cation, quick evaluation of close solutions, and optimization, to name but a few. In this work we show that the classic SA techniques, in particular the continuous sensitivity equation (CSE) method, cannot be used if the mathematical model is a system of hyperbolic partial differential equations (PDEs) with discontinuous solutions. The problem arises from the fact that the CSE method requires the differentiation of the state variable: if the latter is discontinuous, this in turns generates Dirac delta functions in the sensitivity. The focus of the first part of this work is to define a system of sensitivity equations valid also in case of discontinuous state: in order to do that, we add a correction term based on the Rankine-Hugoniot conditions. In the second part of this work we illustrate with some numerical tests how some classical finite volume schemes (an exact Godunov method and a Roe-type method) do not converge to the analytical solution for the sensitivity, due to the numerical diffusion: for this reason, we present an anti-diffusive numerical scheme, which provides the correct results for the sensitivity. In this work we carry out the computation in detail for the barotropic Euler equations in Lagrangian coordinates, but the approach is general and can be applied to any hyperbolic system with discontinuous solutions.

Two forms of continuum shape sensitivity method for fluid–structure interaction problems

Continuum Sensitivity Equation (CSE) methods for deriving and computing derivatives with respect to shape design variables are developed in two forms and compared in their application to fluid–structure interaction (FSI) problems. The local derivative form poses the CSEs in terms of the partial derivatives of the state variables with respect to shape parameters, while the CSEs in total derivative form are posed in terms of the total derivative, also known as the material or substantial derivative. In the literature CSEs are often posed in local form for fluids and total form for solids. The two forms are compared here for the purpose of applying a single form to both fluid and structure domains. The local form, also known as the boundary velocity method, requires design velocity only at the boundaries and interfaces of the domains to pose the CSEs. In contrast, the total form, also known as the domain velocity method, requires the design velocity in the whole domain. The local form requires higher-order spatial derivatives of the analysis solution than the total form, which affects the accuracy of its results. Higher order p-elements are shown to be a remedy to the inaccuracy of local form CSE seen in the literature for finite element solutions. The practicality, accuracy, and efficiency of these two CSE forms are compared based on the implementation and computed derivatives for three examples: a linear Timoshenko beam subject to a tip force, fluid flow around an airfoil, and an airfoil attached to a nonlinear joined beam subject to a gust load.

Continuum Sensitivity Analysis for Structural Shape Design Variables Using Finite-Volume Method

Two of the most important requirements when using gradient-based optimization for fluid–structure interaction problems are efficiency and accuracy in calculating the sensitivities. As a result, analytical continuum sensitivity formulations are finding their place due to their lower cost and accuracy. However, the necessary computation of mesh sensitivities in shape optimization is becoming a bottleneck, especially when handling complex geometries. In this research, an algorithm based on continuum sensitivity analysis for structural shape design variables is developed. In the proposed method, regularized Heaviside functions are used to modify the properties of mesh cells. Therefore, mesh dependency is removed from the sensitivity equations. This method does not require moving or modifying the mesh to handle the shape changes. This method is applied to two different validation cases where continuum sensitivity equations are formulated and solved. Validation cases are selected as a structural and thermal problem to show the applicability of this method for different physics. The results demonstrate the efficiency of the proposed algorithm for design-optimization framework.

Local continuum shape sensitivity with spatial gradient reconstruction

Utilizing gradient-based optimization for large scale, multidisciplinary design problems requires accurate and efficient sensitivity or design derivative analysis. In general, numerical sensitivity methods, such as the finite difference method, are easy to implement but can be computationally expensive and inaccurate. In contrast, analytic sensitivity methods, such as the discrete and continuum methods, are highly accurate but can be very difficult, if not infeasible, to implement. A popular compromise is the semi-analytic method, but it too can be highly inaccurate when computing shape design derivatives. Presented here is an alternative method, which is easy to implement and can be as accurate as conventional analytic sensitivity methods. In this paper a general local continuum shape sensitivity method with spatial gradient reconstruction (SGR) is formulated. It is demonstrated that SGR, a numerical technique, can be used to solve the continuous sensitivity equations (CSEs) in a non-intrusive manner. The method is used to compute design derivatives for a variety of applications, including linear static beam bending, linear transient gust analysis of a 2-D beam structure, linear static bending of rectangular plates, and linear static bending of a beam-stiffened plate. Analysis is conducted with Nastran, and both displacement and stress design derivative solutions are presented. For each example the design derivatives are validated with either analytic or finite difference solutions.

Equivalence of continuum and discrete analytic sensitivity methods for nonlinear differential equations

This paper presents results for the study of equivalence between the total form continuum sensitivity equation (CSE) method and the discrete analytic method of shape design sensitivity analysis. For the discrete analytic method, the sensitivity equations are obtained by taking analytic derivatives of the discretized equilibrium equations with respect to the shape design parameters. For the CSE method, the equilibrium equations are firstly differentiated to form a set of linear continuous sensitivity equations and then discretized for solving the shape sensitivities. The sensitivity equations can be derived by taking the material derivatives (total form) or the partial derivatives (local form) of the equilibrium equations. The total form CSE method is shown for the first time to be equivalent, after finite element discretization, to the discrete analytic method for nonlinear second-order differential equations of a particular form with design dependent loads when they use the same: (1) finite element discretization, (2) numerical integration of element matrices, (3) design velocity fields that are linear with respect to the design variable and (4) shape functions for domain transformation and for design velocity field calculations. The shape sensitivity equations for three-dimensional geometric nonlinear elastic structures and linear potential flow are derived by using both total form CSE and discrete analytic method to show the equivalence of the two methods for these specific examples. The accuracy of shape sensitivity analysis is verified by potential flow around an airfoil and a joined beam with an airfoil under gust load. The results show that analytic sensitivity results are consistent with the complex step results.

Boundary velocity method for continuum shape sensitivity of nonlinear fluid–structure interaction problems

A Continuum Sensitivity Equation (CSE) method was developed in local derivative form for fluid–structure shape design problems. The boundary velocity method was used to derive the continuum sensitivity equations and sensitivity boundary conditions in local derivative form for a built-up joined beam structure under transient aerodynamic loads. For nonlinear problems, when the Newton–Raphson method is used, the tangent stiffness matrix yields the desired sensitivity coefficient matrix for solving the linear sensitivity equations in the Galerkin finite element formulation. For built-up structures with strain discontinuity, the local sensitivity variables are not continuous at the joints, requiring special treatment to assemble the elemental local sensitivities and the generalized force vector. The coupled fluid–structure physics and continuum sensitivity equations for gust response of a nonlinear joined beam with an airfoil model were posed and solved. The results were compared to the results obtained by finite difference (FD) method.

A continuous Lagrangian sensitivity equation method for incompressible flow

A continuous Lagrangian sensitivity equation method (CLSEM) is presented as a cost effective alternative to the continuous (Eulerian) sensitivity equation method (CESEM) in the case of shape parameters. Boundary conditions for the CLSEM are simpler than those of the CESEM. However a mapping must be introduced to relate the undeformed and deformed configurations thus making the PDEs more complicated. We propose the use of pseudo-elasticity equations to provide a general framework to generate this mapping for unstructured meshes on complex geometries. The methodology is presented in details for the incompressible Navier–Stokes and sensitivity equations in variational form. The PDEs are solved with an adaptive FEM. Sensitivity data obtained with both approaches for a flow around a NACA 4512 are used to obtain estimates of flows around nearby geometries. Results indicate that the CLSEM produces significant improvements in terms of both accuracy and CPU time.

MUSCL schemes for the shallow water sensitivity equations with passive scalar transport

A higher-order numerical technique is presented for the direct sensitivity analysis of the shallow water equations with passive scalar transport. The continuous sensitivity equations are modified to account for the possible presence of shocks in the solution, that result in Dirac source terms for the sensitivity across flow discontinuities. Higher-order accuracy is achieved via a MUSCL reconstruction technique with slope limiting, which makes the numerical solution Total Variation Diminishing (TVD). The Harten–Lax–Van Leer (HLL) approximate Riemann solver is modified so as to account for the influence of source terms in both the flow and sensitivity solutions. Several options are tested for the wave speed estimates and the order of the MUSCL time stepping, such as the MUSCL-Hancock, MUSCL-EVR and MUSCL-HLLG techniques. Convergence analyses on continuous and discontinuous flow problems with analytical solutions indicate that first-order time stepping is approximately twice as fast as second-order time stepping and that it yields more accurate sensitivity solutions.

Numerical Noise Reduction with Novel Method of Gradient Calculation

Numerical noise is a prevalent concern in simulation-based optimization problems, where it is usually difficult to exactly describe the objective function. With numerical noise, convergence is not always guaranteed. This paper investigates the use of Continuous Sensitivity Equation (CSE) as the method to calculate the gradient of the inexact objective function in order to reduce numerical noise. Experiments of the problem describing the orbit of a satellite of the earth-moon system with a set of Ordinary Differential Equations (ODEs) and other practical applications in the field of mechanical and electrical engineering are conducted. Gradients are calculated with both CSE and Finite Difference (FE) and compared. CSE yields impressive results for these examples.

Least-Squares Continuous Sensitivity Shape Optimization for Structural Elasticity Applications

A general first-order formulation of the continuous sensitivity equations is introduced that improves the accuracy of the sensitivity boundary conditions. In the continuous sensitivity method, design or shape parameter gradients are computed from a continuous system of partial differential equations instead of the discretized system, which avoids the problematic mesh sensitivity calculations of discrete methods for shape variation problems. The first-order formulation for both the underlying elasticity and sensitivity equations is amenable to solution by a high-order polynomial least-squares finite element model. The continuous sensitivity boundary-value problem, which can be posed in local or total derivative form, is generally simpler in local sensitivity form for shape variation problems. Although the local form is not unique and total material sensitivities are generally desired for structural elasticity problems, the local sensitivity solution is easily transformed to a material derivative. The first-order formulation and the transformation to material derivatives are demonstrated for two elasticity example problems. The least-squares continuous sensitivity solutions are presented and compared with analytic sensitivities and finite difference results and should prove convenient validation benchmarks for other continuous sensitivity applications.

Finite element and sensitivity analysis of thermally induced flow instabilities

AbstractThis paper presents a finite element algorithm for the simulation of thermo‐hydrodynamic instabilities causing manufacturing defects in injection molding of plastic and metal powder. Mold‐filling parameters determine the flow pattern during filling, which in turn influences the quality of the final part. Insufficiently, well‐controlled operating conditions may generate inhomogeneities, empty spaces or unusable parts. An understanding of the flow behavior will enable manufacturers to reduce or even eliminate defects and improve their competitiveness. This work presents a rigorous study using numerical simulation and sensitivity analysis. The problem is modeled by the Navier–Stokes equations, the energy equation and a generalized Newtonian viscosity model. The solution algorithm is applied to a simple flow in a symmetrical gate geometry. This problem exhibits both symmetrical and non‐symmetrical solutions depending on the values taken by flow parameters. Under particular combinations of operating conditions, the flow was stable and symmetric, while some other combinations leading to large thermally induced viscosity gradients produce unstable and asymmetric flow. Based on the numerical results, a stability chart of the flow was established, identifying the boundaries between regions of stable and unstable flow in terms of the Graetz number (ratio of thermal conduction time to the convection time scale) and B, a dimensionless ratio indicating the sensitivity of viscosity to temperature changes. Sensitivities with respect to flow parameters are then computed using the continuous sensitivity equations method. We demonstrate that sensitivities are able to detect the transition between the stable and unstable flow regimes and correctly indicate how parameters should change in order to increase the stability of the flow. Copyright © 2009 John Wiley & Sons, Ltd.

Verified predictions of shape sensitivities in wall-bounded turbulent flows by an adaptive finite-element method

A Continuous Sensitivity Equation (CSE) method is presented for shape parameters in turbulent wall-bounded flows modeled with the standard k– ϵ turbulence model with wall functions. Differentiation of boundary conditions and their complex dependencies on shape parameters, including the two-velocity scale wall functions, is presented in details along with the appropriate methodology required for the CSE method. To ensure accuracy, grid convergence and to reduce computational time, an adaptive finite-element method driven by asymptotically exact error estimations is used. The adaptive process is controlled by error estimates on both flow and sensitivity solutions. Firstly, the proposed approach is applied on a problem with a closed-form solution, derived using the Method of the Manufactured Solution to perform Code Verification. Results from adaptive grid refinement studies show Verification of flow and sensitivity solvers, error estimators and the adaptive strategy. Secondly, we consider turbulent flows around a square cross-section cylinder in proximity of a solid wall. We examine the quality of the numerical solutions by performing Solution Verification and Validation. Then, Sensitivity Analysis of these turbulent flows is performed to investigate the ability of the method to deal with non-trivial geometrical changes. Sensitivity information is used to estimate uncertainties in the flow solution caused by uncertainties in the shape parameter and to perform fast evaluation of flows on nearby configurations.

A parallel computational model for sensitivity analysis in optimization for robustness

We present an efficient method for computing, in a parallel distributed environment, gradients of functionals depending on the solution sensitivities of dynamical systems described by ordinary differential equations. This work was motivated by the need to compute cost function gradients for dynamically constrained optimization for robustness, i.e. the problem of finding the model parameters of a given dynamical system that lead to minimum solution sensitivity. The proposed approach for this particular second-order sensi-tivity problem falls into the class of the so-called ‘adjoint-over-forward’ methods and is based on solving the continuous forward sensitivity equations (for evaluating the cost functional) and their continuous adjoint systems (for evaluation of the gradient of the cost functional). Efficiency is ensured by: (i) a suitable distribution over processors of the individual sensitivity systems, both forward and adjoint so that inter-process communication is minimized; and (ii) use of an iterative solver combined with a block-diagonal preconditioner for the solution of the linear systems arising in the implicit integration of the resulting ordinary differential equations (ODE) systems. The proposed algorithm was implemented as an extension of the cvodes solver in sundials [P. Brown, K. Grant, A. Hindmarsh, S. Lee, R.S.D. Shumaker, and C. Woodward. SUNDIALS: SUite of Nonlinear and DIfferential/ALgebraic equation Solvers, ACM Trans. Math. Software, 31(3) (2005), pp. 363–396], but it can be used in conjunction with any sensitivity-enabled ODE integrator that provides adjoint sensitivity capabilities and (if based on implicit integration methods) support for iterative linear algebra.

Direct numerical simulation of sensitivity coefficients in low Reynolds number turbulent channel flow

The sensitivity equation method (SEM) has been implemented in direct numerical simulations (DNS) of smooth-wall turbulent channel flow to quantify Reynolds number effects on the mean flow field. The present approach towards sensitivity analysis represents a departure from conventional parametric studies wherein multiple numerical simulations are performed with discrete increments in the value of the parameter of interest, in this case the Reynolds number. In the present study, sensitivity derivatives (or sensitivity coefficients) represent the rate of change of the primitive variables with respect to the Reynolds number, and are determined directly by numerically solving the discretized continuous sensitivity equations concurrently with the discretized Navier–Stokes equations. Simulations have been performed at Reynolds numbers of 100 and 180, based on the friction velocity and channel half-width, using a finite volume computational scheme. Wall-normal profiles of the mean streamwise velocity and Reynolds stresses compare very well to others in the literature at the same Reynolds numbers [19,20]. The SEM results correctly predict the expected change in both the mean streamwise velocity and Reynolds shear stress profiles with increasing Reynolds number. Furthermore, the mean SEM results correctly predict the local slope of the skin friction coefficient versus Reynolds number curve, which may have potential use in applications such as boundary layer control by surface modification. The present study is novel insofar as it constitutes the first attempt to implement SEM in an unsteady, fully turbulent, three-dimensional flow field. The additional computational expenses incurred in order to run the SEM simulations in this context are also discussed.

A continuous shape sensitivity equation method for unsteady laminar flows

This paper presents the application of a general shape sensitivity equation method (SEM) to unsteady laminar flows. The formulation accounts for complex parameter dependence and is suitable for a wide range of problems. The flow and sensitivity equations are solved on 3D meshes using a Streamline-Upwind Petrov Galerkin (SUPG) finite element method. In the case of shape parameters, boundary conditions for sensitivities depend on the flow gradient at the boundary. Therefore, an accurate recovery of solution gradients is crucial to the success of shape sensitivity computations. In this work, solution gradients at boundary points are extracted using the Finite Node Displacement (FiND) method on which the finite element discretization is enriched locally via the insertion of nodes close to the boundary points. The normal derivative of the solution is then determined using finite differences. This approach to evaluate shape sensitivity boundary conditions is embedded in the continuous SEM. The methodology is applied to the flow past a cylinder in ground proximity. First, the proposed method is verified on a steady state problem. The computed sensitivity is compared to the actual change in the solution when a small perturbation is imposed to the shape parameter. Then, the study investigates the ability of the SEM to anticipate the unsteady flow response to changes in the ground to cylinder gap. A reduction of the gap causes damping of the vortex shedding while an increase amplifies the unsteadiness.