High-order implicit time integration scheme based on Pad\'e expansions

A single-step high-order implicit time integration scheme for the solution of transient and wave propagation problems is presented. It is constructed from the Pad\'e expansions of the matrix exponential solution of a system of first-order ordinary differential equations formulated in the state-space. A computationally efficient scheme is developed exploiting the techniques of polynomial factorization and partial fractions of rational functions, and by decoupling the solution for the displacement and velocity vectors. An important feature of the novel algorithm is that no direct inversion of the mass matrix is required. From the diagonal Pad\'e expansion of order $M$ a time-stepping scheme of order $2M$ is developed. Here, each elevation of the accuracy by two orders results in an additional system of real or complex sparse equations to be solved. These systems are comparable in complexity to the standard Newmark method, i.e., the effective system matrix is a linear combination of the static stiffness, damping, and mass matrices. It is shown that the second-order scheme is equivalent to Newmark's constant average acceleration method, often also referred to as trapezoidal rule. The proposed time integrator has been implemented in MATLAB using the built-in direct linear equation solvers. In this article, numerical examples featuring nearly one million degrees of freedom are presented. High-accuracy and efficiency in comparison with common second-order time integration schemes are observed. The MATLAB-implementation is available from the authors upon request or from the GitHub repository (to be added).Comment: 43 pages, 19 figure

Quantitative phase analysis by the Rietveld method cement stone hardening on stage

The article presents the results of the quantitative research phase content in the cement stone by the Rietveld method on the curing stage in the range of 6 − 61 hours at temperatures of 40, 50, 70 °C. It was found that the major phases include test cement stone Tobermorite, Deliate, mCanH2O10Si2, where m = 5 or 3 and n = 3 or 1. Phase content was evaluated on the contribution of the individual phases in the integrated intensities calculated, which in turn, is compared with the experimental diffraction pattern. There were also evaluated the mass fractions of phase gratings. Time and temperature isothermal hardening have a significant influence on the structuring of cement stone

Numerical Analysis of a Steam Turbine Rotor subjected to Thermo-Mechanical Cyclic Loads

The contribution at hand discusses the thermo-mechanical analysis of a steam turbine rotor, made of a heat-resistant steel. Thereby, the analysis accounts for the complicated geometry of a real steam turbine rotor, subjected to practical and complex thermo-mechanical boundary conditions. Various thermo-mechanical loading cycles are taken into account, including different starting procedures (cold and warm starts). Within the thermal analysis using the FE code ABAQUS, instationary steam temperatures as well as heat transfer coefficients are prescribed, and the resulting temperature field serves as input for the subsequent structural analysis. In order to describe the mechanical behavior of the heat-resistant steel, which exhibits significant rate-dependent inelasticity combined with hardening and softening phenomena, a robust nonlinear constitutive approach, the binary mixture model, is employed and implemented in ABAQUS in two different ways, i.e. using explicit as well as implicit  methods for the time integration of the governing evolution equations. The numerical performance, the required computational effort, and the obtained accuracy of both integration methods are examined with reference to the thermo-mechanical analysis of a steam turbine rotor, as a typical practical example for the numerical analysis of a complex component. In addition, the obtained temperature, stress, and strain fields in the steam turbine rotor are discussed in detail, and the influence of the different starting procedures is examined closely

Automatic 3D modeling by combining SBFEM and transfinite element shape functions

The scaled boundary finite element method (SBFEM) has recently been employed as an efficient means to model three-dimensional structures, in particular when the geometry is provided as a voxel-based image. To this end, an octree decomposition of the computational domain is deployed and each cubic cell is treated as an SBFEM subdomain. The surfaces of each subdomain are discretized in the finite element sense. We improve on this idea by combining the semi-analytical concept of the SBFEM with certain transition elements on the subdomains' surfaces. Thus, we avoid the triangulation of surfaces employed in previous works and consequently reduce the number of surface elements and degrees of freedom. In addition, these discretizations allow coupling elements of arbitrary order such that local p-refinement can be achieved straightforwardly

Pairing-based algorithms for jacobians of genus 2 curves with maximal endomorphism ring

Using Galois cohomology, Schmoyer characterizes cryptographic non-trivial self-pairings of the $\ell$-Tate pairing in terms of the action of the Frobenius on the $\ell$-torsion of the Jacobian of a genus 2 curve. We apply similar techniques to study the non-degeneracy of the $\ell$-Tate pairing restrained to subgroups of the $\ell$-torsion which are maximal isotropic with respect to the Weil pairing. First, we deduce a criterion to verify whether the jacobian of a genus 2 curve has maximal endomorphism ring. Secondly, we derive a method to construct horizontal $(\ell,\ell)$-isogenies starting from a jacobian with maximal endomorphism ring

High-order implicit time integration scheme with controllable numerical dissipation based on mixed-order Pad\'e expansions

A single-step high-order implicit time integration scheme with controllable numerical dissipation at high frequencies is presented for the transient analysis of structural dynamic problems. The amount of numerical dissipation is controlled by a user-specified value of the spectral radius $\rho_\infty$ in the high frequency limit. Using this user-specified parameter as a weight factor, a Pad\'e expansion of the matrix exponential solution of the equation of motion is constructed by mixing the diagonal and sub-diagonal expansions. An efficient timestepping scheme is designed where systems of equations, similar in complexity to the standard Newmark method, are solved recursively. It is shown that the proposed high-order scheme achieves high-frequency dissipation, while minimizing low-frequency dissipation and period errors. The effectiveness of the provided dissipation control and the efficiency of the scheme are demonstrated by numerical examples. A simple guideline for the choice of the controlling parameter and time step size is provided. The source codes written in MATLAB and FORTRAN are available for download at: https://github.com/ChongminSong/HighOrderTimeIntegration.Comment: 37 pages, 36 figures, 89 equation