Complex variable methods for shape sensitivity of finite element models

Voorhees, Andrew; Millwater, Harry; Bagley, Ronald L.

doi:10.1016/j.finel.2011.05.003

Cited by 47 publications

(28 citation statements)

References 22 publications

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“…The PROM method proposed in the present work is derived using hyper-dual numbers , which are based on a branch of complex number theory. A similar approach based on ordinary complex numbers (Voorhees et al, 2011) has shown some success already. The inclusion of hyper-dual numbers into the model development process allows for the exact calculations of derivatives (as opposed to approximations from using ordinary complex numbers, which are subject to numerical error, as discussed in the following text and in Fig.…”

Section: Hyper-dual Number Parameterizationsmentioning

confidence: 99%

“…Given a generalized complex number E, the complex step derivative calculation method (Abreu et al, 2013;Voorhees et al, 2011) is defined from the Taylor series of a function f (x) expanded about a step hE, where h typically is a small, real-valued number…”

Section: Dual Numbers Hyper-dual Numbers and The Complex Step Derivmentioning

confidence: 99%

“…reduced order model (PROM) of the system (Hong et al, 2012(Hong et al, , 2013Kammer and Nimityongskul, 2011;Kammer and Krattiger, 2013;Momin et al, 2010;Voorhees et al, 2011). This allows the behavior of the system to be predicted inexpensively over a range of perturbations based on a set of simulations distributed about the nominal design.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Parameterized reduced order models from a single mesh using hyper-dual numbers

Brake¹,

Fike²,

Topping

2016

Journal of Sound and Vibration

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In order to assess the predicted performance of a manufactured system, analysts must consider random variations (both geometric and material) in the development of a model, instead of a single deterministic model of an idealized geometry with idealized material properties. The incorporation of random geometric variations, however, potentially could necessitate the development of thousands of nearly identical solid geometries that must be meshed and separately analyzed, which would require an impractical number of man-hours to complete. This research advances a recent

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Section: Hyper-dual Number Parameterizationsmentioning

confidence: 99%

Section: Dual Numbers Hyper-dual Numbers and The Complex Step Derivmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Parameterized reduced order models from a single mesh using hyper-dual numbers

Brake¹,

Fike²,

Topping

2016

Journal of Sound and Vibration

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“…Since then it has been used in a number of computational fluid dynamics solvers [10,15], nonlinear structural analysis [4], pseudo-spectral simulations [1], dynamical systems [3], fatigue analysis [17] and shape sensitivity [16].…”

Section: Introductionmentioning

confidence: 99%

Numerical Sensitivity Computation for Discontinuous Gradient-Only Optimization Problems Using the Complex-Step Method

Wilke¹,

Kok²

2014

Proceedings of 10th World Congress on Computational Mechanics

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Abstract. This study considers the numerical sensitivity calculation for discontinuous gradientonly optimization problems using the complex-step method. The complex-step method was initially introduced to differentiate analytical functions in the late 1960s, and is based on a Taylor series expansion using a pure imaginary step. The complex-step method is not subject to subtraction errors as with finite difference approaches when computing first order sensitivities and therefore allows for much smaller step sizes that ultimately yields accurate sensitivities. This study investigates the applicability of the complex-step method to numerically compute first order sensitivity information for discontinuous optimization problems. An attractive feature of the complex-step approach is that no real difference step is taken as with conventional finite difference approaches, since conventional finite differences are problematic when real steps are taken over a discontinuity. We highlight the benefits and disadvantages of the complex-step method in the context of discontinuous gradient-only optimization problems that result from numerically approximated (partial) differential equations.Gradient-only optimization is a recently proposed alternative to mathematical programming for solving discontinuous optimization problems. Gradient-only optimization was initially proposed to solve shape optimization problems that utilise remeshing (i.e. the mesh topology is allowed to change) between design updates. Here, changes in mesh topology result in abrupt changes in the discretization error of the computed response. These abrupt changes in turn manifests as discontinuities in the numerically computed objective and constraint functions of an optimization problem. These discontinuities are in particular problematic when they manifest as local minima. Note that these potential issues are not limited to problems in shape optimization but may be present whenever (partial) differential equations are approximated numerically with non-constant discretization methods e.g. remeshing of spatial domains or automatic time stepping in temporal domains.

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“…All computations were performed using the aforementioned finite element beam formulation written in MATLAB. The prescribed boundary conditions are given in Eqs (11,12). were used in the Newmark-beta method.These values in the literature have shown to give exceptional accuracy of the solution when !…”

mentioning

confidence: 99%

Sensitivity Analysis in Structural Dynamics using the ZFEM Complex Variable Finite Element Method

Garza

Millwater²

2013

54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

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SPECIAL SESSION ON THE DIGITAL TWINSensitivity anal 1 ysis of structural systems is of great importance for structural dynamic modifications. The sensitivity analysis helps identify key input parameters that influence the dynamic response of a model. The present paper demonstrates how to accurately and efficiently obtain derivatives of linear dynamic systems using the complex step method and the generalized multicomplex step method implemented within a complex variable finite element method (ZFEM). The highly accurate derivatives are computed using a single complex variable finite element analysis of the structural system. An undamped and damped simply supported beam was modeled using ZFEM and the response derivatives were computed with respect to load amplitude, load frequency, beam cross sectional dimension, and Rayleigh damping coefficients. Comparisons of the derivatives against analytical and finite difference estimates verify the accuracy of the methodology. Nomenclatureth undamped natural frequency ! " n = n th mode shape ! K e = local stiffness matrix ! M e = local consistent mass matrix ! " = mass-normalized mode shape matrix ! " = diagonal matrix of natural frequencies squared ! " = vector of nodal degrees of freedom ! I = identity matrix ! "t = time step ! " = forcing frequency ! Q = uniform load ! " = Rayleigh damping mass proportional coefficient ! " = Rayleigh damping stiffness proportional coefficient ! * = denotes a complex variable

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Complex variable methods for shape sensitivity of finite element models

Cited by 47 publications

References 22 publications

Parameterized reduced order models from a single mesh using hyper-dual numbers

Parameterized reduced order models from a single mesh using hyper-dual numbers

Numerical Sensitivity Computation for Discontinuous Gradient-Only Optimization Problems Using the Complex-Step Method

Sensitivity Analysis in Structural Dynamics using the ZFEM Complex Variable Finite Element Method

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