Approximate Jacobian Research Articles

Analyzing the permeability distribution of multilayered specimens using pulsed eddy-current testing with multi-scale 1D-ResNet

The mechanical properties of steel are determined by its microstructure, which is closely related to its permeability profile. In thermal processing, layered structures are formed in steel and different layers have different mechanical and magnetic properties. Therefore, it is crucial to propose a practical method to monitor the change of permeability profile along the depth, which can indicate the evolution of the microstructure of steel during thermal processing, such as hot rolling. This paper presents a method for determining the layered structure and permeability profile of the steel by using pulsed eddy current testing (PECT), which offers better penetration ability. An analytical model has been deduced for calculating the time-domain pulsed eddy current (PEC) response from a Hall sensor of a triple-layer conductor system based on the inverse Laplace transform. It is found the Tau (τ) curve is closely related to the permeability profile of the conductor. For the inverse solution, the Simultaneous Iterative Reconstruction Technique (SIRT) is utilized to determine the permeability profile of the multilayered specimens from the measured response. The approximate Jacobian matrix (sensitivity matrix) is obtained by the perturbation method based on the Tau curve. However, the permeability profile suffers from smoothing effect and sharp features are lost. Deep learning (DL) algorithm based on the Multi-Scale 1D-ResNet model is therefore introduced to address this issue. Numerical simulations and experiments have been performed to evaluate the proposed method for permeability profile estimation with various materials and thicknesses. The DL method can achieve an accurate estimation of the plate permeability profile with a relative error under 5%.

MATRICS: The implicit matrix-free Eulerian hydrodynamics solver

Context. There exists a zoo of different astrophysical fluid dynamics solvers, most of which are based on an explicit formulation and hence stability-limited to small time steps dictated by the Courant number expressing the local speed of sound. With this limitation, the modeling of low-Mach-number flows requires small time steps that introduce significant numerical diffusion, and a large amount of computational resources are needed. On the other hand, implicit methods are often developed to exclusively model the fully incompressible or 1D case since they require the construction and solution of one or more large (non)linear systems per time step, for which direct matrix inversion procedures become unacceptably slow in two or more dimensions. Aims. In this work, we present a globally implicit 3D axisymmetric Eulerian solver for the compressible Navier–Stokes equations including the energy equation using conservative formulation and a fully simultaneous approach. We use the second-order-in-time backward differentiation formula for temporal discretization as well as the κ scheme for spatial discretization. We implement different limiter functions to prohibit the occurrence of spurious oscillations in the vicinity of discontinuities. Our method resembles the well-known monotone upwind scheme for conservation laws (MUSCL). We briefly present efficient solution methods for the arising sparse and nonlinear system of equations. Methods. To deal with the nonlinearity of the Navier–Stokes equations we used a Newton iteration procedure in which the required Jacobian matrix-vector product was reconstructed with a first-order finite difference approximation to machine precision in a matrix-free way. The resulting linear system was solved either completely matrix-free with a combination of a sufficient Krylov solver and an approximate Jacobian preconditioner or semi-matrix-free with the incomplete lower upper factorization technique as a preconditioner. The latter was dependent on a standalone approximation of the Jacobian matrix, which was optionally calculated and needed solely for the purpose of preconditioning. Results. We show our method to be capable of damping sound waves and resolving shocks even at Courant numbers larger than one. Furthermore, we prove the method’s ability to solve boundary value problems like the cylindrical Taylor-Couette flow (TC), including viscosity, and to model transition flows. To show the latter, we recover predicted growth rates for the vertical shear instability, while choosing a time step orders of magnitude larger than the explicit one. Finally, we verify that our method is second order in space by simulating a simplistic, stationary solar wind.

A comparison of Rosenbrock–Wanner and Crank–Nicolson time integrators for atmospheric modelling

AbstractNonhydrostatic atmospheric models often use semi‐implicit temporal discretisations in order to negate the time‐step limitation of explicitly resolving the fast acoustic and gravity waves. Solving the resulting system to machine precision using Newton's method is considered prohibitively expensive, and so the nonlinear solver is typically truncated to a fixed number of iterations, often using an approximate Jacobian matrix that is reassembled only once per time step. The present article studies the impact of using various third‐order, four stage Rosenbrock–Wanner schemes, where integration weights are chosen to meet specific stability and order conditions, in comparison to a Crank–Nicolson time discretisation, as is done in the UK Met Office's LFRic model. Rosenbrock–Wanner schemes present a promising alternative on account of their ability to preserve their temporal order with only an approximate Jacobian, and may be constructed to be stiffly‐stable, so as to ensure the decay of fast unresolved modes. These schemes are compared for the 2D rotating shallow‐water equations and the 3D compressible Euler equations at both planetary and nonhydrostatic scales and are shown to exhibit improved results in terms of their energetic profiles and stability. Results in terms of computational performance are mixed, with the Crank–Nicolson method allowing for longer time steps and faster time to solution for the baroclinic instability test case at planetary scales, and the Rosenbrock–Wanner methods allowing for longer time steps and faster time to solution for a rising bubble test case at nonhydrostatic scales.

Retrieving 3D distributions of atmospheric particles using Atmospheric Tomography with 3D Radiative Transfer – Part 2: Local optimization

Abstract. Our global understanding of clouds and aerosols relies on the remote sensing of their optical, microphysical, and macrophysical properties using, in part, scattered solar radiation. Current retrievals assume clouds and aerosols form plane-parallel, homogeneous layers and utilize 1D radiative transfer (RT) models. These assumptions limit the detail that can be retrieved about the 3D variability in the cloud and aerosol fields and induce biases in the retrieved properties for highly heterogeneous structures such as cumulus clouds and smoke plumes. In Part 1 of this two-part study, we validated a tomographic method that utilizes multi-angle passive imagery to retrieve 3D distributions of species using 3D RT to overcome these issues. That validation characterized the uncertainty in the approximate Jacobian used in the tomographic retrieval over a wide range of atmospheric and surface conditions for several horizontal boundary conditions. Here, in Part 2, we test the algorithm's effectiveness on synthetic data to test whether the retrieval accuracy is limited by the use of the approximate Jacobian. We retrieve 3D distributions of a volume extinction coefficient (σ3D) at 40 m resolution from synthetic multi-angle, mono-spectral imagery at 35 m resolution derived from stochastically generated cumuliform-type clouds in (1 km)3 domains. The retrievals are idealized in that we neglect forward-modelling and instrumental errors, with the exception of radiometric noise; thus, reported retrieval errors are the lower bounds. σ3D is retrieved with, on average, a relative root mean square error (RRMSE) < 20 % and bias < 0.1 % for clouds with maximum optical depth (MOD) < 17, and the RRMSE of the radiances is < 0.5 %, indicating very high accuracy in shallow cumulus conditions. As the MOD of the clouds increases to 80, the RRMSE and biases in σ3D worsen to 60 % and −35 %, respectively, and the RRMSE of the radiances reaches 16 %, indicating incomplete convergence. This is expected from the increasing ill-conditioning of the inverse problem with the decreasing mean free path predicted by RT theory and discussed in detail in Part 1. We tested retrievals that use a forward model that is not only less ill-conditioned (in terms of condition number) but also less accurate, due to more aggressive delta-M scaling. This reduces the radiance RRMSE to 9 % and the bias in σ3D to −8 % in clouds with MOD ∼ 80, with no improvement in the RRMSE of σ3D. This illustrates a significant sensitivity of the retrieval to the numerical configuration of the RT model which, at least in our circumstances, improves the retrieval accuracy. All of these ensemble-averaged results are robust in response to the inclusion of radiometric noise during the retrieval. However, individual realizations can have large deviations of up to 18 % in the mean extinction in clouds with MOD ∼ 80, which indicates large uncertainties in the retrievals in the optically thick limit. Using less ill-conditioned forward model tomography can also accurately infer optical depths (ODs) in conditions spanning the majority of oceanic cumulus fields (MOD < 80), as the retrieval provides ODs with bias and RRMSE values better than −8 % and 36 %, respectively. This is a significant improvement over retrievals using 1D RT, which have OD biases between −30 % and −23 % and RRMSE between 29 % and 80 % for the clouds used here. Prior information or other sources of information will be required to improve the RRMSE of σ3D in the optically thick limit, where the RRMSE is shown to have a strong spatial structure that varies with the solar and viewing geometry.

Accelerated nonlinear finite element method for analysis of isotropic hyperelastic materials nonlinear deformations

In this work, an approximate Jacobian matrix is proposed based on the total Lagrangian formulation of Finite Element Method for isotropic hyperelastic materials. The approximate Jacobian matrix can take the place of the exact Jacobian matrix in the Newton–Raphson method to avoid frequent construction and factorization of the Jacobian matrix. A new Quasi–Newton method employing the approximate Jacobian matrix is developed to significantly improve the convergence rate. The proposed method was tested on three examples with the combinations of three hyperelastic material models and three element types. The results show that the proposed method is more efficient than ABAQUS and CALCULIX (an open-source FEM software package) on all of the tests without loss of accuracy. It is up to 100 times faster than the traditional Quasi–Newton method, and at least 2.5 times faster than ABAQUS.

Retrieving 3D distributions of atmospheric particles using Atmospheric Tomography with 3D Radiative Transfer – Part 1: Model description and Jacobian calculation

Abstract. Our global understanding of clouds and aerosols relies on the remote sensing of their optical, microphysical, and macrophysical properties using, in part, scattered solar radiation. These retrievals assume that clouds and aerosols form plane-parallel, homogeneous layers and utilize 1D radiative transfer (RT) models, limiting the detail that can be retrieved about the 3D variability in cloud and aerosol fields and inducing biases in the retrieved properties for highly heterogeneous structures such as cumulus clouds and smoke plumes. To overcome these limitations, we introduce and validate an algorithm for retrieving the 3D optical or microphysical properties of atmospheric particles using multi-angle, multi-pixel radiances and a 3D RT model. The retrieval software, which we have made publicly available, is called Atmospheric Tomography with 3D Radiative Transfer (AT3D). It uses an iterative, local optimization technique to solve a generalized least squares problem and thereby find a best-fitting atmospheric state. The iterative retrieval uses a fast, approximate Jacobian calculation, which we have extended from Levis et al. (2020) to accommodate open and periodic horizontal boundary conditions (BCs) and an improved treatment of non-black surfaces. We validated the accuracy of the approximate Jacobian calculation for derivatives with respect to both the 3D volume extinction coefficient and the parameters controlling the open horizontal boundary conditions across media with a range of optical depths and single-scattering properties and find that it is highly accurate for a majority of cloud and aerosol fields over oceanic surfaces. Relative root mean square errors in the approximate Jacobian for a 3D volume extinction coefficient in media with cloud-like single-scattering properties increase from 2 % to 12 % as the maximum optical depths (MODs) of the medium increase from 0.2 to 100.0 over surfaces with Lambertian albedos <0.2. Over surfaces with albedos of 0.7, these errors increase to 20 %. Errors in the approximate Jacobian for the optimization of open horizontal boundary conditions exceed 50 %, unless the plane-parallel media providing the boundary conditions are optically very thin (∼0.1). We use the theory of linear inverse RT to provide insight into the physical processes that control the cloud tomography problem and identify its limitations, supported by numerical experiments. We show that the Jacobian matrix becomes increasing ill-posed as the optical size of the medium increases and the forward-scattering peak of the phase function decreases. This suggests that tomographic retrievals of clouds will become increasingly difficult as clouds become optically thicker. Retrievals of asymptotically thick clouds will likely require other sources of information to be successful. In Loveridge et al. (2023a; hereafter Part 2), we examine how the accuracy of the retrieved 3D volume extinction coefficient varies as the optical size of the target medium increases using synthetic data. We do this to explore how the increasing error in the approximate Jacobian and the increasingly ill-posed nature of the inversion in the optically thick limit affect the retrieval. We also assess the accuracy of retrieved optical depths and compare them to retrievals using 1D radiative transfer.

A spectral MUSCL scheme for gPC-Galerkin method to uncertain hyperbolic equations

This paper deals with a problem that we found when directly implementing the scalar MUSCL scheme to spectral coefficient equations of the gPC-Galerkin method. The order degradation at extrema of spectral coefficients and non-uniform reconstruction process lead to a distortion of standard deviation at some extrema, especially where the peak consists of several extrema of coefficients. From the perspective of probability space, a spectral MUSCL scheme is proposed based on the generalized minmod limiter to avoid the defects mentioned above when applying the scalar MUSCL. In this paper, we present some properties of the new scheme related to total variation and demonstrate them by the uncertain linear scalar conservation law. Finally, we employ this scheme to compressible Euler equations, and a good correlation of standard deviation is obtained. The details of implementation of the gPC-Galerkin method for Euler equations are also discussed, and the approximate Jacobian matrix is adopted.

On the effect of nonlinearity and Jacobian initialization on the convergence of the generalized Broyden quasi‐Newton method

AbstractThis article investigates two aspects of the generalized Broyden quasi‐Newton method that have a major impact on its convergence: the initial approximation of the Jacobian and the presence of nonlinearities in the secant conditions. After reformulating the common representation of generalized Broyden, a straightforward interpretation is given. This leads to a natural extension of the method in which an application‐dependent physics‐based surrogate model is used as initial approximation of the (inverse) Jacobian. A carefully chosen surrogate has the potential to greatly reduce the required number of iterations. The behavior of generalized Broyden depends strongly on the parameter that determines how many secant conditions are satisfied by the Jacobian approximation. Respecting all secant conditions reduces it to Anderson acceleration; a single one to Broyden's original method. An analysis demonstrates that these two variants behave very differently when nonlinearities are present in the secant conditions: they are ignored by Broyden, but can destabilize Anderson. On the other hand, the analysis shows that Broyden tends to neglect small linear information, possibly reducing convergence speed. To mitigate stability problems with Anderson acceleration, a practical method to detect and remove nonlinear secant information is introduced next. Finally, we solve a steady free‐surface‐flow problem using several generalized Broyden variants, testing the influence of the surrogate, the nonlinearities and the combination thereof. The results agree with the theoretical predictions, showing large differences in convergence behavior. Furthermore, the proposed method effectively negates the problems related to nonlinearities in this case.

Education

The Education section in this issue presents two contributions. The first paper, “$LU$ and $CR$ Elimination,” is written by Gilbert Strang and Cleve Moler. This is an expository paper which is helpful to instructors and students interested in linear algebra and matrix manipulations. The focus of the paper is the column-row factorization for any matrix $A$ of rank $r$. The matrix is represented as $A=CR$, where the matrix $C$ contains the first $r$ independent columns of $A$, and the matrix $R$ contains the nonzero rows of the reduced row echelon form of $A$. The authors explain how to obtain the factorization and contrast it with $LU$, $QR$ factorizations, the single-value decomposition, and another factorization, $A= CW^{-1}B$, which they call magic. In section 6, the authors point to an important timely application pertaining to randomized algorithms for a low-rank approximation of large sparse matrices. The paper is clearly and nicely written. It includes a discussion about relevant MATLAB functions, examples, and several references to randomized linear algebra literature. The second paper is “Newton's Method in Mixed Precision,” presented by C. T. Kelley. The paper discusses the impact of precision in calculations on the rate of convergence of Newton's method for solving a system of nonlinear equations $F(x)=0.$ The vector-function $F$ is differentiable and its Jacobian $F' (x)$ is nonsingular. It is assumed that the step $s$ in each iteration is determined by solving the linear equation $F' (x)s= - F(x)$ via Gaussian elimination with column pivoting. The paper starts with the classical result on the local convergence of the method under standard assumptions and then passes onto analyzing the effect of errors arising in calculation of the function and in approximation of the Jacobian. The author's derivations predict no significant difference in the convergence of the nonlinear iteration between a double-precision analytic Jacobian with double precision in the linear solver and a forward-difference approximate Jacobian with a single precision in the linear solver. An interesting part of the paper deals with the question of estimating the effect of the backward error in the solver. The author observes that the worst case estimates are too pessimistic and rarely seen in practice. Invoking recent techniques in probabilistic rounding analysis, he derives more realistic results. The theoretical statements are supplemented by a numerical illustration in section 3. An additional example demonstrates how the theory breaks down if the Jacobian is singular at the solution.

NURBS-Diff: A Differentiable Programming Module for NURBS

Boundary representations (B-reps) using Non-Uniform Rational B-splines (NURBS) are the de facto standard used in CAD, but their utility in deep learning-based approaches is not well researched. We propose a differentiable NURBS module to integrate NURBS representations of CAD models with deep learning methods. We mathematically define the derivatives of the NURBS curves or surfaces with respect to the input parameters (control points, weights, and the knot vector). These derivatives are used to define an approximate Jacobian used for performing the “backward” evaluation to train the deep learning models. We have implemented our NURBS module using GPU-accelerated algorithms and integrated it with PyTorch, a popular deep learning framework. We demonstrate the efficacy of our NURBS module in performing CAD operations such as curve or surface fitting and surface offsetting. Further, we show its utility in deep learning for unsupervised point cloud reconstruction and enforce analysis constraints. These examples show that our module performs better for certain deep learning frameworks and can be directly integrated with any deep-learning framework requiring NURBS.

Finite-time adaptive control for the dual-arm space robots with uncertain kinematics, dynamics and deadzone nonlinearities

Dual-arm space robots are capable of load transporting and coordinated manipulation for on-orbit servicing. However, achieving the accurate trajectory tracking performance is a big challenge for dual-arm robots, especially when mechanical system uncertainties exist. This paper proposes an adaptive control scheme for the dual-arm space robots with grasped targets to accurately follow trajectories while stabilizing base’s attitude in the presence of dynamic uncertainties, kinematic uncertainties and deadzone nonlinearities. An approximate Jacobian matrix is utilized to compensate the kinematic uncertainties, while a radial basis function neural network (RBFNN) with feature decomposition technique is employed to approximate the unknown dynamics. Besides, a smooth deadzone inverse is introduced to reduce the effects from deadzone nonlinearities. The adaption laws for the parameters of the approximate Jacobian matrix, RBFNN and the deadzone inverse are designed with the consideration of the finite-time convergence of trajectory tracking errors as well as the parameters estimation. The stability of the control scheme is validated by a defined Lyapunov function. Several simulations were conducted, and the simulation results verified the effectiveness of the proposed control scheme.

A Quasi‐Newton Reformulated Geostatistical Approach on Reduced Dimensions for Large‐Dimensional Inverse Problems

AbstractFor classic geostatistical approaches relying on Gauss‐Newton or quasilinear methods, iterative evaluations of forward models to compute the Jacobians is a major computational challenge for large‐dimensional, underdetermined nonlinear inverse problems. Recent developments in dimension reduction based on principal component analysis have lower the number of forward model implementations in each iteration to the number of retained principal components given prior covariance or structural parameters. The present research develops a quasi‐Newton approach on reduced dimensions within the reformulated geostatistical approach framework to further lower the number of forward model calls and thus decrease the overall computational cost. The proposed algorithm uses Broyden's method to update the approximate Jacobian of the forward model with respect to the projections on the retained principal axes and the drift coefficients. When the secant condition is satisfied, no additional forward model runs or adjoint solver runs are needed. The efficiency and validity of the developed approach are tested by numerical experiments of large‐dimensional hydraulic tomography with true or biased prior information. Compared with Newton method using the full‐rank update of the Jacobians, the quasi‐Newton reformulated geostatistical approach provides similarly high‐quality inverse estimates with a 60–70% reduction of required forward model simulations. For inverse problems with biased prior information, the iterative correction of the principal axes or the update of the prior covariance may be influenced by the rank‐one update of the Jacobian, yielding slightly lower map accuracies, but still providing satisfactory inverse results with substantially reduced computational costs.

Shamanskii Method for Solving Parameterized Fuzzy Nonlinear Equations

One of the most significant problems in fuzzy set theory is solving fuzzy nonlinear equations. Numerous researches have been done on numerical methods for solving these problems, but numerical investigation indicates that most of the methods are computationally expensive due to computing and storage of Jacobian or approximate Jacobian at every iteration. This paper presents the Shamanskii algorithm, a variant of Newton method for solving nonlinear equation with fuzzy variables. The algorithm begins with Newton’s step at first iteration, followed by several Chord steps thereby reducing the high cost of Jacobian or approximate Jacobian evaluation during the iteration process. The fuzzy coe?cients of the nonlinear systems are parameterized before applying the proposed algorithm to obtain their solutions. Preliminary results of some benchmark problems and comparisons with existing methods show that the proposed method is promising.

Perturbation and numerical solutions of non‐Newtonian fluid bounded within in a porous channel: Applications of pseudo‐spectral collocation method

In the current study, the effects of fluidic parameters with entropy generation properties on velocity, temperature, and entropy numbers of non-Newtonian fluid flowing through the porous channel are investigated. The complex system of fluid equations is handled with analytically and numerically. The first-order perturbation expansion is employed on both velocity and temperature to obtain the approximate analytical solution. A comparison of the analytical solution is made with numerical results that are obtained by discretizing the system of boundary value problems. The pseudo-spectral collocation method was used for the discretization, and the Newton method was to get the solutions to the complex differential equations. In the Newton method, the finite difference approximation of Jacobian is utilized. The pseudo-spectral solutions are in good agreement with the analytical findings. The order of accuracy in temperature and velocity profiles is of order 10−6 which will be compared in the future with the experimental results of given non-Newtonian fluid.

A computationally-efficient method for flamelet calculations

A new open-source code for the simulation of the diffusion flamelet equations is proposed. Emphasis is placed on using an approximate Jacobian to reduce the computational cost of the matrix operations. Performance of the proposed solvers is tested by performing flamelet calculations with kinetic mechanisms of varying sizes. For the unity Lewis number equations, the present iterative Newton solver using an approximate Jacobian greatly outperforms direct Newton solvers using exact Jacobians. The computation cost scales linearly with the number of species, leading to a reduction in solution times by two orders of magnitude for mechanisms containing thousands of species. The applicability of the Jacobian approximations to the solution of the non-unity Lewis number flamelet equations is assessed. The approximations are generally inadequate to solve the full non-unity Lewis number equations but can be used in some applications depending on the balance of terms in the flamelet equations. As an example, the flamelet solver is applied to the study of sooting tendencies in laminar co-flow diffusion flames where modified non-unity Lewis number flamelet equations, previously shown to accurately reproduce experimentally-measured Yield Sooting Indices (YSI), are solved. The accelerated flamelet solver is well suited for sensitivity analysis and uncertainty quantification with large detailed kinetic mechanisms, tasks for which the computational cost was previously prohibitive.

An efficient quasi‐Newton method for two‐dimensional steady free surface flow

SummarySteady free surface flows are of interest in the fields of marine and hydraulic engineering. Fitting methods are generally used to represent the free surface position with a deforming grid. Existing fitting methods tend to use time‐stepping schemes, which is inefficient for steady flows. There also exists a steady iterative method, but that one needs to be implemented with a dedicated solver. Therefore a new method is proposed to efficiently simulate two‐dimensional (2D) steady free surface flows, suitable for use in conjunction with black‐box flow solvers. The free surface position is calculated with a quasi‐Newton method, where the approximate Jacobian is constructed in a novel way by combining data from past iterations with an analytical model based on a perturbation analysis of a potential flow. The method is tested on two 2D cases: the flow over a bottom topography and the flow over a hydrofoil. For all simulations the new method converges exponentially and in few iterations. Furthermore, convergence is independent of the free surface mesh size for all tests.

Nonintrusive continuum sensitivity analysis for fluid applications

Continuum Sensitivity Analysis (CSA) provides an analytic method of computing derivatives for structures, fluids and fluid-structure-interaction problems with respect to shape or value parameters. CSA does not require mesh sensitivity to calculate local shape sensitivity, thereby contributing to its computational efficiency. CSA involves solving linear sensitivity equations with the corresponding sensitivity boundary conditions. Spatial gradients of flow variables are required to construct the sensitivity boundary conditions, the accuracy of which significantly affects the flow shape derivatives. A method called Spatial Gradient Reconstruction (SGR) is used to accurately compute the flow spatial gradients that are further used to set the sensitivity boundary conditions. Here we present a nonintrusive CSA formulation for calculating the material derivatives of flow variables with respect to shape design parameters. Nonintrusive CSA means CSA without any modification to the black-box program used for the CFD flow analysis. Moreover, nonintrusive CSA is flow solver agnostic, in that, knowledge of the solver's discretization is not needed. An example of two-dimensional flow over a NACA0012 airfoil demonstrates the application of nonintrusive CSA to steady flow problems involving Euler (compressible inviscid) equations over unstructured grids with finite volume spatial discretization. Accuracy was studied by investigating convergence for a family of six high-quality grids, ranging from four thousand to four million cells. Factors such as the accuracy of the flow spatial gradients, flow sensitivity boundary conditions, weak implementation of the boundary conditions in the finite volume framework, and the use of an approximate flux Jacobian matrix, all of which affect the accuracy of the material derivatives, are discussed. A novelty of this work is that the sensitivity analysis is done nonintrusively using FLUENT and SU2 software to establish the use of black-box codes for obtaining flow derivatives using the CSA approach.

A time-spectral approximate Jacobian based linearized compressible Navier-Stokes solver for high-speed boundary-layer receptivity and stability

A numerical method for conducting linear receptivity and stability investigations of high-speed wall-bounded flows based on the linearized compressible Navier-Stokes equations is presented. The current approach is directly applicable for stability investigations of arbitrarily complex geometries. The left-hand-side operator for the linear system of equations is obtained by computing numerical right-hand-side Jacobians while the right-hand-side is build from exact flux Jacobians. Utilizing the numerical right-hand-side Jacobian approach avoids lengthy, error prone derivation of the stability equations in the context of generalized curvilinear coordinates. The governing equations are solved using either time-stepping or time-spectral discretizations. Three different time-spectral approaches, i.e., direct inversion, unfactored and factored schemes, are presented and their numerical characteristics for the solution of the linearized Navier-Stokes equations for linear receptivity and stability analysis for large-scale transition problems are explored. Linear receptivity and stability calculation results are provided for different solver options. Performance comparison of the three schemes are presented for a wide range of test cases: An incompressible cross-flow for a swept flat plate boundary layer, a supersonic shock-boundary-layer interaction, hypersonic boundary layers on a flat plate and a flared cone, and, finally, for the receptivity of a hypersonic boundary layer for a right sharp cone.

A Jacobian-free approximate Newton–Krylov startup strategy for RANS simulations

The favorable convergence rates of Newton–Krylov-based solution algorithms have increased their popularity for computational fluid dynamics applications. Unfortunately, these methods perform poorly during the initial stages of convergence, particularly for three-dimensional Reynolds-averaged Navier–Stokes simulations. Addressing this problem requires the use of a globalization method such as pseudo-transient continuation, along with an approximate Newton–Krylov startup stage. This class of methods marches the solution in pseudo-time with a matrix-based approximate Jacobian that has a lower bandwidth and better conditioning properties compared with the exact Jacobian. However, this matrix-based approach also has shortcomings, including the large cost of computing and storing the approximate Jacobian and its preconditioner, along with the need for an updated Jacobian for every nonlinear iteration. To rectify these shortcomings, we use approximate residual formulations in a Jacobian-free approximate Newton–Krylov algorithm. With the approximate Jacobian, we compute the vector products by using the approximate residual computations in a matrix-free manner while forming a preconditioner based on the matrix-based approximate Jacobian. This approach keeps the approximate Jacobian up to date and mitigates the cost of forming a matrix-based Jacobian and its preconditioner at each iteration by lagging the preconditioner between nonlinear iterations. We use varying levels of approximations with the matrix-free approach and thereby demonstrate the trade-off between rate of convergence and the cost of each nonlinear iteration. The proposed implementation uses only the exact and approximate residual formulations and can therefore be generalized with minimal additional implementation effort to a range of solvers and discretizations. The code is available under an open-source license.

Sparse, iterative simulation methods for one-dimensional laminar flames

Sparse, iterative simulation methods for one-dimensional laminar flames are proposed. The resulting steady and unsteady flame solvers exploit approximate Jacobians to greatly reduce the computational cost associated with matrix operations. The constant, non-unity Lewis number assumption is introduced to further reduce the computational cost. The solvers are also parallelized to reduce time-to-solution on distributed memory computer systems. Computed laminar flame speeds and species profiles for a range of chemical mechanisms (from 10 to 2878 species) are compared against a well-validated commercial code and are found to be consistent within solver tolerances. The computation times of both the unsteady and steady solutions increase only linearly with the number of species, which is a significant improvement over the quadratic or cubic scaling of existing steady-state flame solvers. For the largest mechanism tested, the steady-state flame solver is two orders of magnitude faster than commonly-used codes. The use of an approximate Jacobian is shown to reduce the rate of convergence for the steady-state solver, but does not significantly affect the domain of convergence. The steady-state solver with approximate Jacobian is thus well suited for computationally efficient laminar flame speed sweeps with large kinetic mechanisms.