On evaluating higher-order derivatives of the QR decomposition of tall matrices with full column rank in forward and reverse mode algorithmic differentiation

2012 ◽  
Vol 27 (2) ◽  
pp. 391-403 ◽  
Author(s):  
Sebastian F. Walter ◽  
Lutz Lehmann ◽  
René Lamour
Keyword(s):  
Qr Decomposition ◽  
Higher Order ◽  
Full Column Rank ◽  
Algorithmic Differentiation ◽  
Reverse Mode ◽  
Derivatives Of

Reverse-mode algorithmic differentiation of an OpenMP-parallel compressible flow solver

2017 ◽  
Vol 33 (1) ◽  
pp. 140-154 ◽  
Author(s):  
Jan Hückelheim ◽  
Paul Hovland ◽  
Michelle Mills Strout ◽  
Jens-Dominik Müller
Keyword(s):  
Compressible Flow ◽  
Finite Volume ◽  
High Performance ◽  
Computational Cost ◽  
Design Parameters ◽  
Flow Solver ◽  
Algorithmic Differentiation ◽  
Reverse Mode ◽  
Many Core ◽  
Derivatives Of

Reverse-mode algorithmic differentiation (AD) is an established method for obtaining adjoint derivatives of computer simulation applications. In computational fluid dynamics (CFD), adjoint derivatives of a cost function output such as drag or lift with respect to design parameters such as surface coordinates or geometry control points are a key ingredient for shape optimization, uncertainty quantification and flow control. The computational cost of CFD applications and their derivatives makes it essential to use high-performance computing hardware efficiently, including multi- and many-core architectures. Nevertheless, OpenMP is not supported in most AD tools, and previously shown methods achieve poor scalability of the derivative code. We present the AD of an OpenMP-parallelized finite volume compressible flow solver for unstructured meshes. Our approach enables us to reuse the parallelization of the original code in the computation of adjoint derivatives. The method works by identifying code segments that can be differentiated in reverse-mode without changing their memory access pattern. The OpenMP parallelization is integrated into the derivative code during the build process in a way that is robust to modifications of the original code and independent of the OpenMP support of the differentiation tool. We show the scalability of our adjoint CFD solver on test cases ranging from thousands to millions of finite volume mesh cells on CPUs with up to 16 threads as well as on an Intel XeonPhi card with 236 threads. We demonstrate that our approach is more practical to implement for production-sized CFD codes and produces more efficient adjoint derivative code than previously shown AD methods.

A C0 three-node triangular element based on preprocessing approach for thick sandwich plates

2017 ◽  
Vol 21 (6) ◽  
pp. 1820-1842
Author(s):  
Wu Zhen ◽  
Ma Rui ◽  
Chen Wanji
Keyword(s):  
Transverse Shear ◽  
Equilibrium Equation ◽  
Three Dimensional ◽  
Higher Order ◽  
Triangular Element ◽  
Shear Stresses ◽  
Sandwich Plates ◽  
Transverse Shear Stress ◽  
Transverse Shear Stresses ◽  
Derivatives Of

This paper will try to overcome two difficulties encountered by the C0 three-node triangular element based on the displacement-based higher-order models. They are (i) transverse shear stresses computed from constitutive equations vanish at the clamped edges, and (ii) it is difficult to accurately produce the transverse shear stresses even using the integration of the three-dimensional equilibrium equation. Invalidation of the equilibrium equation approach ought to attribute to the higher-order derivations of displacement parameters involved in transverse shear stress components after integrating three-dimensional equilibrium equation. Thus, the higher-order derivatives of displacement parameters will be taken out from transverse shear stress field by using the three-field Hu–Washizu variational principle before the finite element procedure is implemented. Therefore, such method is named as the preprocessing method for transverse shear stresses in present work. Because the higher-order derivatives of displacement parameters have been eliminated, a C0 three-node triangular element based on the higher-order zig-zag theory can be presented by using the linear interpolation function. Performance of the proposed element is numerically evaluated by analyzing multilayered sandwich plates with different loading conditions, lamination sequences, material constants and boundary conditions, and it can be found that the present model works well in the finite element framework.

Expansions of the mildly relativistic dispersion function for nearly perpendicular electron cyclotron ray tracing

1999 ◽  
Vol 61 (1) ◽  
pp. 121-128 ◽  
Author(s):  
I. P. SHKAROFSKY
Keyword(s):  
Ray Tracing ◽  
Computer Programs ◽  
Higher Order ◽  
Electron Cyclotron ◽  
Dispersion Function ◽  
Relativistic Dispersion ◽  
Plasma Dispersion ◽  
Derivatives Of ◽  
Cyclotron Harmonic ◽  
Experienced Difficulty

To trace rays very close to the nth electron cyclotron harmonic, we need the mildly relativistic plasma dispersion function and its higher-order derivatives. Expressions for these functions have been obtained as an expansion for nearly perpendicular propagation in a region where computer programs have previously experienced difficulty in accuracy, namely when the magnitude of (c/vt)2 (ω−nωc)/ω is between 1 and 10. In this region, the large-argument expansions are not yet valid, but partial cancellations of terms occur. The expansion is expressed as a sum over derivatives of the ordinary dispersion function Z. New expressions are derived to relate higher-order derivatives of Z to Z itself in this region of concern in terms of a finite series.

Forced Response Sensitivity Analysis Using an Adjoint Harmonic Balance Solver

2018 ◽  
Author(s):  
Anna Engels-Putzka ◽  
Jan Backhaus ◽  
Christian Frey
Keyword(s):  
Frequency Domain ◽  
Harmonic Balance ◽  
Unsteady Aerodynamics ◽  
Harmonic Balance Method ◽  
Forced Response ◽  
Algorithmic Differentiation ◽  
Design And Optimization ◽  
Initial Application ◽  
Reverse Mode ◽  
Geometric Changes

This paper describes the development and initial application of an adjoint harmonic balance solver. The harmonic balance method is a numerical method formulated in the frequency domain which is particularly suitable for the simulation of periodic unsteady flow phenomena in turbomachinery. Successful applications of this method include unsteady aerodynamics as well as aeroacoustics and aeroelasticity. Here we focus on forced response due to the interaction of neighboring blade rows. In the CFD-based design and optimization of turbomachinery components it is often helpful to be able to compute not only the objective values — e.g. performance data of a component — themselves, but also their sensitivities with respect to variations of the geometry. An efficient way to compute such sensitivities for a large number of geometric changes is the application of the adjoint method. While this is frequently used in the context of steady CFD, it becomes prohibitively expensive for unsteady simulations in the time domain. For unsteady methods in the frequency domain, the use of adjoint solvers is feasible, but still challenging. The present approach employs the reverse mode of algorithmic differentiation (AD) to construct a discrete adjoint of an existing harmonic balance solver in the framework of an industrially applied CFD code. The paper discusses implemen-tational issues as well as the performance of the adjoint solver, in particular regarding memory requirements. The presented method is applied to compute the sensitivities of aeroelastic objectives with respect to geometric changes in a turbine stage.

Open Higher Order Continuous-Time Dynamic Model with Mixed Stock and Flow Data and Derivatives of Exogenous Variables

1991 ◽  
Vol 7 (3) ◽  
pp. 404-408 ◽  
Author(s):  
K. Ben Nowman
Keyword(s):  
Continuous Time ◽  
Higher Order ◽  
Flow Data ◽  
Continuous Time Model ◽  
Exogenous Variables ◽  
Time Model ◽  
Time Dynamic ◽  
Mixed Stock ◽  
Gaussian Estimation ◽  
Derivatives Of

This paper is concerned with deriving formulae for higher order derivatives of exogenous variables for use in estimating the parameters of an open secondorder continuous time model with mixed stock and flow data and first and second order derivatives of exogenous variables which are not observable. This should provide the basis for the future estimation of continuous time models in a range of applied areas using the new Gaussian estimation computer program developed by Nowman [4].

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