Lixiang Yang

Ultrasonic backscattering in polycrystals with elongated single phase and duplex microstructures

An analytical solution for a three dimensional integral representation of the backscattering (BS) coefficient in polycrystals with elongated (generally ellipsoidal) grains is obtained; it is a natural generalization of the known explicit result for the BS coefficient in polycrystals with spherical grains. New insights into the dependence of the BS signal on frequency and averaged ellipsoidal grain radii are obtained. In particular it has been shown that the dominant factor for the backscattering is the averaged interaction length of the ellipsoidal grain in the direction of wave propagation, instead of the ellipsoidal cross-section. The theory was applied to a simplified model of Ti alloy duplex microstructure and was compared with experiment. For the experimental data analysis directional backscattering ratios are introduced and shown to be advantageous for characterization of duplex elongated microstructures/microtextures. In addition to the geometrical parameters of the elongated microtextures, the BS directional ratios depend on the newly introduced nondimensional material parameter q. The parameter q exhibits the relative contribution of the second phase (crystallites) to the backscattering signal, the effect of which is measurable and important. Comparison of the model with experiment shows there is a significant advantage in using the directional ratios of backscattering coefficients for data analysis.

Effect of texture and grain shape on ultrasonic backscattering in polycrystals.

An ultrasonic backscattering model is developed for textured polycrystalline materials with orthotropic or trigonal grains of ellipsoidal shape. The model allows us to simulate realistic microstructures and orthotropic macroscopic material textures resulting from thermomechanical processing for a broad variety of material symmetries. The 3-D texture is described by a modified Gaussian orientation distribution function (ODF) of the crystallographic orientation of the grains along the macroscopic texture direction. The preferred texture directions are arbitrary relative to the axes of the ellipsoidal grains. The averaged elastic covariance and the directional anisotropy of the backscattering coefficient are obtained for a wave propagation direction arbitrary relative to the texture and grain elongation directions. One particular application of this analysis is the backscattering solution for cubic crystallites with common textures such as Cube, Goss, Brass and Copper. In our analysis, in the texture-defined coordinates the matrix of elastic constants for cubic crystallites takes the form of orthotropic or trigonal symmetry. Numerical results are presented, discussed and compared to the experimental data available in the literature illustrating the dependence of the backscattering coefficient on texture and grain shape.

Velocity-Stress Equations for Waves in Solids of Hexagonal Symmetry Solved by the Space-Time CESE Method

This paper reports an extension of the space-time conservation element and solution element (CESE) method to simulate stress waves in elastic solids of hexagonal symmetry. The governing equations include the equation of motion and the constitutive equation of elasticity. With velocity and stress components as the unknowns, the governing equations are a set of 9, first-order, hyperbolic partial differential equations. To assess numerical accuracy of the results, the characteristic form of the equations is derived. Moreover, without using the assumed plane wave solution, the one-dimensional equations are shown to be equivalent to the Christoffel equations. The CESE method is employed to solve an integral form of the governing equations. Space-time flux conservation over conservation elements (CEs) is imposed. The integration is aided by the prescribed discretization of the unknowns in each solution element (SE), which in general does not coincide with a CE. To demonstrate this approach, numerical results in the present paper include one-dimensional expansion waves in a suddenly stopped rod, two-dimensional wave expansion from a point in a plane, and waves interacting with interfaces separating hexagonal solids with different orientations. All results show salient features of wave propagation in hexagonal solids and the results compared well with the available analytical solutions.

Numerical Solution by the CESE Method of a First-Order Hyperbolic Form of the Equations of Dynamic Nonlinear Elasticity

This paper reports the application of the space-time conservation element and solution element (CESE) method to the numerical solution of nonlinear waves in elastic solids. The governing equations consist of a pair of coupled first-order nonlinear hyperbolic partial differential equations, formulated in the Eulerian frame. We report their derivations and present conservative, nonconservative, and diagonal forms. The conservative form is solved numerically by the CESE method; the other forms are used to study the eigenstructure of the hyperbolic system (which reveals the underlying wave physics) and deduce the Riemann invariants. The proposed theoretical/numerical approach is demonstrated by directly solving two benchmark elastic wave problems: one involving linear propagating extensional waves, the other involving nonlinear resonant standing waves. For the extensional wave problem, the CESE method accurately captures the sharp propagating wavefront without excessive numerical diffusion or spurious oscillations, and predicts correct reflection characteristics at the boundaries. For the resonant vibrations problem, the CESE method captures the linear-to-nonlinear evolution of the resonant waves and the distribution of wave energy among multiple modes in the nonlinear regime.

SOLVCON: A Python-Based CFD Software Framework for Hybrid Parallelization

SOLVCON is a new, open-sourced software framework for high-fidelity solutions of linear and non-linear hyperbolic partial differential equations. SOLVCON emphasizes scalability, portability, and maintainability for supercomputing by using emerging multi-core architectures. The code development effort follows Extreme Programming practices, including version control, documentation, issue tracking, user support, and frequent code releases. In SOLVCON, the Python framework includes all supportive functionalities for the work flow. For pre-processing operations, the Python framework provides parallelized mesh data input and automatically sets up domain decomposition. In calculations, the Python framework provides light-weight memory management through extensive use of pointers. Computation-intensive operations are implemented by using C and FORTRAN for high performance. The default numerical algorithm employed is the space-time Conservation Element and Solution Element (CESE) method. The code uses general unstructured meshes with mixed elements, including tetrahedra, hexahedra, prisms, and pyramids for threedimensional calculations. Hybrid parallelism includes shared- and distributed-memory parallelization. The temporal loop and the spatial loop in modern finite-volume methods are implemented in a two-layered structure in SOLVCON. Distributed-memory parallelization by domain decomposition and MPI is performed in the temporal loop. Shared-memory parallel computing by using accelerator technologies, e.g., General-Purpose Graphic Processor Unit (GPGPU), is performed in the spatial loop. More than 99% of the execution time of SOLVCON is used for number-crunching in the solver as a part of the space loop. Written in C or FORTRAN, a typical solver contains only 10% of the code statements in SOLVCON. To demonstrate the capabilities of newly developed SOLVCON, we performed CFD calculations by using 23 million elements. The code was run on a 512-core cluster. SOLVCON delivers calculations of flow variables in 11.29 million elements per second. The parallel efficiency is 70%. In the open-sourced SOLVCON, two solvers are available: (i) the Euler equations solver for compressible flows, and (ii) the velocity-stress equations solver for waves in anisotropic elastic solids. SOLVCON can be easily extended for other applications, including viscous flows, aero-acoustics, nonlinear solid mechanics, and electromagnetism. The Python framework allows fast adaption to new heterogeneous, multi-core hardware as well as further development of the code for peta-scale supercomputing.

Eigenstructure of First-Order Velocity-Stress Equations for Waves in Elastic Solids of Trigonal 32 Symmetry

This paper reports the eigenstructure of a set of first-order hyperbolic partial differential equations for modeling waves in solids with a trigonal 32 symmetry. The governing equations include the equation of motion and partial differentiation of the elastic constitutive relation with respect to time. The result is a set of nine, first-order, fully coupled, hyperbolic partial differential equations with velocity and stress components as the unknowns. Shown in the vector form, the model equations have three 9×9 coefficient matrices. The wave physics are fully described by the eigenvalues and eigenvectors of these matrices; i.e., the nontrivial eigenvalues are the wave speeds, and a part of the corresponding left eigenvectors represents wave polarization. For a wave moving in a certain direction, three wave speeds can be identified by calculating the eigenvalues of the coefficient matrix in a rotated coordinate system. In this process, without using the plane-wave solution, we recover the Christoffel matrix and thus validate the formulation. To demonstrate this approach, two- and three-dimensional slowness profiles of quartz are calculated. Wave polarization vectors for wave propagation in several compression directions as well as noncompression directions are discussed.

Velocity-Stress Equations for Wave Propagation in Anisotropic Elastic Media

where ρ is the density of the medium, w the displacement, and c[4] the fourth-order stiffness tensor [2]. Equation (1.1) has been derived based on the equation of motion in conjunction with the elastic constitutive equation. Equation (1.1) has been solved by the finite-difference methods, e.g., [21], and the time-domain finite-element methods, e.g., [33], for propagating waves, and the frequency-domain finite-element methods, e.g., [6], for normal mode analysis of standing waves.