Kirchhoff Theory

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Thomas J. R. Hughes - One of the best experts on this subject based on the ideXlab platform.

  • generalization of the twist Kirchhoff Theory of plate elements to arbitrary quadrilaterals and assessment of convergence
    Computer Methods in Applied Mechanics and Engineering, 2012
    Co-Authors: H A F A Santos, John A. Evans, Thomas J. R. Hughes
    Abstract:

    Abstract We generalize the recently introduced twist-Kirchhoff Theory of rectangular plate elements to arbitrary quadrilateral elements. A key feature is the use of Raviart–Thomas vector-field approximations for rotations. To preserve continuity of the normal components of the rotation vector across mesh edges, we employ the Piola transformation to map the rotations from the parent domain to the physical domain. These elements possess a unique combination of efficiency and robustness in that minimal quadrature rules are sufficient to guarantee stability without rank deficiency. In particular, only one-point Gauss quadrature is required for the lowest-order element in the twist-Kirchhoff family. We numerically study the convergence and accuracy of the first two members of the twist-Kirchhoff family of quadrilateral elements on square, rhombic and circular plate problems.

  • New rectangular plate elements based on twist-Kirchhoff Theory
    Computer Methods in Applied Mechanics and Engineering, 2011
    Co-Authors: Franco Brezzi, John A. Evans, Thomas J. R. Hughes, Luisa Donatella Marini
    Abstract:

    Abstract We introduce a new framework for the development of thin plate finite elements, the “twist-Kirchhoff Theory”. A family of rectangular plate elements is derived that takes advantage of the special structure of this new Theory. Particular attention is focused on the lowest-order member of the family, an eight degree-of-freedom, four-node element with mid-side rotations whose stiffness matrix is exactly computed with one-point Gaussian quadrature. We prove a convergence theorem for it and various error estimates. These are also generalized to the higher-order elements in the family. Numerical tests corroborate the theoretical results.

Richard V. Craster - One of the best experts on this subject based on the ideXlab platform.

  • diffuse scattered field of elastic waves from randomly rough surfaces using an analytical Kirchhoff Theory
    Journal of The Mechanics and Physics of Solids, 2016
    Co-Authors: Michael J. S. Lowe, X Xi, Richard V. Craster
    Abstract:

    Abstract We develop an elastodynamic Theory to predict the diffuse scattered field of elastic waves by randomly rough surfaces, for the first time, with the aid of the Kirchhoff approximation (KA). Analytical expressions are derived incorporating surface statistics, to represent the expectation of the angular distribution of the diffuse intensity for different modes. The analytical solutions are successfully verified with numerical Monte Carlo simulations, and also validated by comparison with experiments. We then apply the Theory to quantitatively investigate the effects of the roughness and the shear-to-compressional wave speed ratio on the mode conversion and the scattering intensity, from low to high roughness within the valid region of KA. Both the direct and the mode converted intensities are significantly affected by the roughness, which leads to distinct scattering patterns for different wave modes. The mode conversion effect is very strong around the specular angle and it is found to increase as the surface appears to be more rough. In addition, the 3D roughness induced coupling between the out-of-plane shear horizontal (SH) mode and the in-plane modes is studied. The intensity of the SH mode is shown to be very sensitive to the out-of-plane correlation length, being influenced more by this than by the RMS value of the roughness. However, it is found that the depolarization pattern for the diffuse field is independent of the actual value of the roughness.

  • The validity of Kirchhoff Theory for scattering of elastic waves from rough surfaces
    Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences, 2015
    Co-Authors: Fan Shi, Wonjae Choi, Michael J. S. Lowe, E.a. Skelton, Richard V. Craster
    Abstract:

    The Kirchhoff approximation (KA) for elastic wave scattering from two-dimensional (2D) and three-dimensional (3D) rough surfaces is critically examined using finite-element (FE) simulations capable of extracting highly accurate data while retaining a fine-scale rough surface. The FE approach efficiently couples a time domain FE solver with a boundary integration method to compute the scattered signals from specific realizations of rough surfaces. Multiple random rough surfaces whose profiles have Gaussian statistics are studied by both Kirchhoff and FE models and the results are compared; Monte Carlo simulations are used to assess the comparison statistically. The comparison focuses on the averaged peak amplitude of the scattered signals, as it is an important characteristic measured in experiments. Comparisons, in both two dimensions and three dimensions, determine the accuracy of Kirchhoff Theory in terms of an empirically estimated parameter σ 2 /λ 0 ( σ is the RMS value, and λ 0 is the correlation length, of the roughness), being considered accurate when this is less than some upper bound c , ( σ 2 /λ 0 c ). The incidence and scattering angles also play important roles in the validity of the Kirchhoff Theory and it is found that for modest incidence angles of less than 30°, the accuracy of the KA is improved even when σ 2 /λ 0 > c . In addition, the evaluation results are compared using 3D isotropic rough surfaces and 2D surfaces with the same surface parameters.

Dominik Prazak - One of the best experts on this subject based on the ideXlab platform.

Miloslav Ohlídal - One of the best experts on this subject based on the ideXlab platform.

Michael J. S. Lowe - One of the best experts on this subject based on the ideXlab platform.

  • diffuse scattered field of elastic waves from randomly rough surfaces using an analytical Kirchhoff Theory
    Journal of The Mechanics and Physics of Solids, 2016
    Co-Authors: Michael J. S. Lowe, X Xi, Richard V. Craster
    Abstract:

    Abstract We develop an elastodynamic Theory to predict the diffuse scattered field of elastic waves by randomly rough surfaces, for the first time, with the aid of the Kirchhoff approximation (KA). Analytical expressions are derived incorporating surface statistics, to represent the expectation of the angular distribution of the diffuse intensity for different modes. The analytical solutions are successfully verified with numerical Monte Carlo simulations, and also validated by comparison with experiments. We then apply the Theory to quantitatively investigate the effects of the roughness and the shear-to-compressional wave speed ratio on the mode conversion and the scattering intensity, from low to high roughness within the valid region of KA. Both the direct and the mode converted intensities are significantly affected by the roughness, which leads to distinct scattering patterns for different wave modes. The mode conversion effect is very strong around the specular angle and it is found to increase as the surface appears to be more rough. In addition, the 3D roughness induced coupling between the out-of-plane shear horizontal (SH) mode and the in-plane modes is studied. The intensity of the SH mode is shown to be very sensitive to the out-of-plane correlation length, being influenced more by this than by the RMS value of the roughness. However, it is found that the depolarization pattern for the diffuse field is independent of the actual value of the roughness.

  • The validity of Kirchhoff Theory for scattering of elastic waves from rough surfaces
    Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences, 2015
    Co-Authors: Fan Shi, Wonjae Choi, Michael J. S. Lowe, E.a. Skelton, Richard V. Craster
    Abstract:

    The Kirchhoff approximation (KA) for elastic wave scattering from two-dimensional (2D) and three-dimensional (3D) rough surfaces is critically examined using finite-element (FE) simulations capable of extracting highly accurate data while retaining a fine-scale rough surface. The FE approach efficiently couples a time domain FE solver with a boundary integration method to compute the scattered signals from specific realizations of rough surfaces. Multiple random rough surfaces whose profiles have Gaussian statistics are studied by both Kirchhoff and FE models and the results are compared; Monte Carlo simulations are used to assess the comparison statistically. The comparison focuses on the averaged peak amplitude of the scattered signals, as it is an important characteristic measured in experiments. Comparisons, in both two dimensions and three dimensions, determine the accuracy of Kirchhoff Theory in terms of an empirically estimated parameter σ 2 /λ 0 ( σ is the RMS value, and λ 0 is the correlation length, of the roughness), being considered accurate when this is less than some upper bound c , ( σ 2 /λ 0 c ). The incidence and scattering angles also play important roles in the validity of the Kirchhoff Theory and it is found that for modest incidence angles of less than 30°, the accuracy of the KA is improved even when σ 2 /λ 0 > c . In addition, the evaluation results are compared using 3D isotropic rough surfaces and 2D surfaces with the same surface parameters.

  • Improved detection of rough defects for ultrasonic nondestructive evaluation inspections based on finite element modeling of elastic wave scattering
    IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control, 2015
    Co-Authors: James R. Pettit, Anthony E. Walker, Michael J. S. Lowe
    Abstract:

    Defects which possess rough surfaces greatly affect ultrasonic wave scattering behavior, usually reducing the magnitude of reflected signals. Understanding and accurately predicting the influence of roughness on signal amplitudes is crucial, especially in nondestructive evaluation (NDE) for the inspection of safety-critical components. An extension of Kirchhoff Theory has formed the basis for many practical applications; however, it is widely recognized that these predictions are pessimistic because of analytical approximations. A numerical full-field modeling approach does not fall victim to such limitations. Here, a finite element (FE) modeling approach is used to develop a realistic methodology for the prediction of expected backscattering from rough defects. The ultrasonic backscatter from multiple rough surfaces defined by the same statistical class is calculated for normal and oblique incidence. Results from FE models are compared with Kirchhoff Theory predictions and experimental measurements to establish confidence in the new approach. At lower levels of roughness, excellent agreement is observed between Kirchhoff Theory, FE, and experimental data, whereas at higher values, the pessimism of Kirchhoff Theory is confirmed. An important distinction is made between the total, coherent, and diffuse signals and it is observed, significantly, that the total signal amplitude is representative of the information obtained during an inspection. This analysis provides a robust basis for a less sensitive, yet safe, threshold for inspection of rough defects.

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