A stiffness derivative local hypercomplex-variable finite element method for computing the energy release rate

Aguirre-Mesa, Andres M.; Ramirez-Tamayo, Daniel; García, Manuel J.; Montoya, Arturo; Millwater, Harry

doi:10.1016/j.engfracmech.2019.106581

Cited by 11 publications

(4 citation statements)

References 34 publications

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“…To build the complex system of equations the Cauchy–Riemann matrix representation stipulates that the real part of the function is located across the matrix diagonal 11 so that the independent contributions are

{\underset{{\overset{K}{true}}_{italicuu}}{false}}^{*} = [\begin{matrix} Re \{{\underline{K_{uu}}}^{*}\} & - Im \{{\underline{K_{uu}}}^{*}\} \\ Im \{{\underline{K_{uu}}}^{*}\} & Re \{{\underline{K_{uu}}}^{*}\} \end{matrix}],

{\underset{{\overset{K}{true}}_{italicuv}}{false}}^{*} = [\begin{matrix} Re \{{\underline{K_{uv}}}^{*}\} & - Im \{{\underline{K_{uv}}}^{*}\} \\ Im \{{\underline{K_{uv}}}^{*}\} & Re \{{\underline{K_{uv}}}^{*}\} \end{matrix}],

{\underset{{\overset{K}{true}}_{italicvv}}{false}}^{*} = [\begin{matrix} Re \{{\underline{K_{vv}}}^{*}\} & - Im \{{\underline{K_{vv}}}^{*}\} \\ Im \{{\underline{K_{vv}}}^{*}\} & Re \{...\} \end{matrix}]

…”

Section: Imaginary Finite Element Implementationmentioning

confidence: 99%

Sensitivity analysis of a layered piezoelectric system using ZFEM

Acosta,

Bhalla,

Guo

2023

Int J Numerical Modelling

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The complex variable finite element method (ZFEM) is a numerical technique which aims to find the partial derivatives of the independent variables with respect to variation in dependent parameters declared in the physics. This is done by combining the complex Taylor series expansion within the weak formulation of the governing equation in a coupled system of linear equations forming a complex valued block matrix given by the Cauchy–Riemann matrix representation. In this work, two‐dimensional linear first‐order elements have been implemented in ZFEM to predict the design derivatives of the mechanical displacement field and the voltage potential field for a layered piezoelectric system in a steady‐state study with Dirichlet boundary condition applied at the top and bottom edges of the geometry. This approach allows the standard FEM solution to quantify the sensitivity of the mechanical displacement and voltage potential fields with respect to small variations in the material properties through the information obtained from the computation of the derivatives. The domain is formed by a layered body with PZT‐4 and PZT‐5 stacked together. For result verification, the numerical solution obtained with ZFEM was compared to results from a commercial FEM package and the solution from the imaginary part was compared to the exact solution of a well‐known benchmark problem. Comparison of the results showed good agreement for both the real and imaginary parts of the solution and the largest sensitivities were found in PZT‐5 specifically in , , and .

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{\underset{{\overset{K}{true}}_{italicuu}}{false}}^{*} = [\begin{matrix} Re \{{\underline{K_{uu}}}^{*}\} & - Im \{{\underline{K_{uu}}}^{*}\} \\ Im \{{\underline{K_{uu}}}^{*}\} & Re \{{\underline{K_{uu}}}^{*}\} \end{matrix}],

{\underset{{\overset{K}{true}}_{italicuv}}{false}}^{*} = [\begin{matrix} Re \{{\underline{K_{uv}}}^{*}\} & - Im \{{\underline{K_{uv}}}^{*}\} \\ Im \{{\underline{K_{uv}}}^{*}\} & Re \{{\underline{K_{uv}}}^{*}\} \end{matrix}],

{\underset{{\overset{K}{true}}_{italicvv}}{false}}^{*} = [\begin{matrix} Re \{{\underline{K_{vv}}}^{*}\} & - Im \{{\underline{K_{vv}}}^{*}\} \\ Im \{{\underline{K_{vv}}}^{*}\} & Re \{...\} \end{matrix}]

…”

Section: Imaginary Finite Element Implementationmentioning

confidence: 99%

Sensitivity analysis of a layered piezoelectric system using ZFEM

Acosta,

Bhalla,

Guo

2023

Int J Numerical Modelling

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“…( 6) has no subtraction cancellation error, similar to the complex-step derivative approximation in Eq. (4). It also has no truncation error and therefore has no dependence on the step size h. In practice, a step size of h ¼ 1 is chosen so that h can be omitted from Eq.…”

Section: Iia First-order Partial Derivative Expressionsmentioning

confidence: 99%

“…For example, hypercomplex algebra has been implemented in finite element analysis (FEA) to calculate second-order derivatives of the stiffness matrix for energy release rate sensitivities. 4 A second-order Kalman filter using multicomplex numbers was developed in Ref. 5.…”

Section: Introductionmentioning

confidence: 99%

Multidual Sensitivity Method in Ray-Tracing Transport Simulations

Balcer

Millwater

Favorite

2021

Nuclear Science and Engineering

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The multidual differentiation method has been implemented in a ray-tracing transport simulation for the purpose of calculating arbitrary-order sensitivities of the uncollided particle leakage. This method extends dual number differentiation by perturbing variables along multiple nonreal axes to calculate arbitrary-order derivatives. Numerical results of first-through third-order multidual sensitivities of the uncollided particle leakage with respect to isotope densities, microscopic cross sections, source emission rates, and material interface locations (including the outer boundary) are shown for a two-region sphere. The relative error of first and second partial derivatives with respect to isotopic parameters and first partial derivatives of the leakage with respect to interface locations are within 9.8E−10% of existing adjoint-based sensitivities. Higher-order multidual-based derivatives that are not available with the adjoint method are in excellent agreement with central difference approximations.

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“…The accuracy depends on effective meshing and sensitivity of the crack front shape and smaller crack growth increments. The virtual crack closure technique (VCCT) option offers a simpler but more general way for obtaining the energy release rate estimation in crack propagation at the crack tip which requires additional post-processing procedures with more precise results [8,9]. Rybicki and Kanninen [10] proposed VCCT in two-dimensional finite element stress intensity analysis with a coarse grid for modes I and II.…”

Section: Introductionmentioning

confidence: 99%

Numerical Evaluation of Fatigue Crack Growth of Structural Steels Using Energy Release Rate with VCCT

et al. 2022

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This research presents the numerical evaluation of fatigue crack growth of structural steels S355 and S960 based on Paris’ law parameters (C and m) that are experimentally determined with a single edge notched tension (SENT) specimen using optical and crack gauge measurements on an electromotive resonance machine at constant amplitude load. The sustainable technique is replacing destructive, time-consuming and expensive approaches in structural integrity. The crack propagation is modelled using the 3D finite element method (FEM) with adaptive remeshing of tetrahedral elements along with the crack initiator elements provided in simulation software for crack propagation based on linear elastic fracture mechanics (LEFM). The stress intensity is computed based on the evaluation of energy release rates according to Irwin’s crack closure integral with applied cyclic load of 62.5 MPa, 100 MPa and 150 MPa and stress ratios of R = 0 and 0.1. In order to achieve optimized mesh size towards load cycle and computational time, mesh and re-mesh sensitivity analysis is conducted. The results indicate that the virtual crack closure technique VCCT-based 3D FEM shows acceptable agreement compared to the experimental investigation with the percentage error up to 7.9% for S355 and 12.8% for S960 structural steel.

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A stiffness derivative local hypercomplex-variable finite element method for computing the energy release rate

Cited by 11 publications

References 34 publications

Sensitivity analysis of a layered piezoelectric system using ZFEM

Sensitivity analysis of a layered piezoelectric system using ZFEM

Multidual Sensitivity Method in Ray-Tracing Transport Simulations

Numerical Evaluation of Fatigue Crack Growth of Structural Steels Using Energy Release Rate with VCCT

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