3D Numerical Prediction of Thermal Weakening of Granite under Tension

Geosciences ◽  
2021 ◽  
Vol 12 (1) ◽  
pp. 10
Author(s):  
Timo Saksala
Keyword(s):  
Tensile Strength ◽  
Rock Fracture ◽  
Critical Time ◽  
Heterogeneous Material ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Numerical Approach ◽  
Numerical Prediction ◽  
Time Step ◽  
Mass Scaling

This paper deals with numerical prediction of temperature (weakening) effects on the tensile strength of granitic rock. A 3D numerical approach based on the embedded discontinuity finite elements is developed for this purpose. The governing thermo-mechanical initial/boundary value problem is solved with an explicit (in time) staggered method while using extreme mass scaling to increase the critical time step. Rock fracture is represented by the embedded discontinuity concept implemented here with the linear (4-node) tetrahedral elements. The rock is modelled as a linear elastic (up to fracture by the Rankine criterion) heterogeneous material consisting of Quartz, Feldspar and Biotite minerals. Due to its strong and anomalous temperature dependence upon approaching the α-β transition at the Curie point (~573 °C), only Quartz in the numerical rock depends on temperature in the present approach. In the numerical testing, the sample is first volumetrically heated to a target temperature. Then, the uniaxial tension test is performed on the cooled down sample. The simulations demonstrate the validity of the proposed approach as the experimental deterioration, by thermally induced cracking, of the rock tensile strength is predicted with a good accuracy.

scholarly journals Numerical Modeling of Temperature Effect on Tensile Strength of Granitic Rock

2021 ◽  
Vol 11 (10) ◽  
pp. 4407
Author(s):  
Timo Saksala
Keyword(s):  
Tensile Strength ◽  
Temperature Effect ◽  
Curie Point ◽  
Granitic Rock ◽  
Critical Time ◽  
Slow Heating ◽  
Single Element ◽  
Target Temperature ◽  
Time Step ◽  
Mass Scaling

The aim of this paper is to numerically predict the temperature effect on the tensile strength of granitic rock. To this end, a numerical approach based on the embedded discontinuity finite elements is developed. The underlying thermo-mechanical problem is solved with a staggered method marching explicitly in time while using extreme mass scaling, allowed by the quasi-static nature of the slow heating of a rock sample to a uniform target temperature, to increase the critical time step. Linear triangle elements are used to implement the embedded discontinuity kinematics with two intersecting cracks in a single element. It is assumed that the quartz mineral, with its strong and anomalous temperature dependence upon approaching the α-β transition at the Curie point (~573 °C), in granitic rock is the major factor resulting in thermal cracking and the consequent degradation of tensile strength. Accordingly, only the thermal expansion coefficient of quartz depends on temperature in the present approach. Moreover, numerically, the rock is taken as isotropic except for the tensile strength, which is unique for each mineral in a rock. In the numerical simulations mimicking the experimental setup on granitic numerical rock samples consisting of quartz, feldspar and biotite minerals, the sample is first heated slowly to a target temperature below the Curie point. Then, a uniaxial tension test is numerically performed on the cooled down sample. The simulations demonstrate the validity of the proposed approach as the experimental deterioration of the tensile strength of the rock is predicted with agreeable accuracy.

Increasing the critical time step: micro-inertia, inertia penalties and mass scaling

2011 ◽  
Vol 47 (6) ◽  
pp. 657-667 ◽  
Author(s):  
Harm Askes ◽  
Duc C. D. Nguyen ◽  
Andy Tyas
Keyword(s):  
Critical Time ◽  
Time Step ◽  
Mass Scaling ◽  
Critical Time Step

scholarly journals A difference scheme with the optimal weight for the diffusion-convection equation

2019 ◽  
pp. 283-292
Author(s):  
А.И. Сухинов ◽  
А.Е. Чистяков ◽  
В.В. Сидорякина ◽  
С.В. Проценко
Keyword(s):  
Difference Scheme ◽  
Approximation Error ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Step Function ◽  
Time Step ◽  
Optimal Weight ◽  
Convection Equation ◽  
Weight Value ◽  
Diffusion Convection

Исследована разностная схема с весами для однородного пространственно-одномерного уравнения диффузии-конвекции. Выполнено исследование погрешности аппроксимации разностной схемы в зависимости от шага по времени на основе разложения функции решения и погрешности аппроксимации по тригонометрическому базису. Разработан алгоритм нахождения оптимального значения веса, обеспечивающий минимум погрешности аппроксимации решения исходной начально-краевой задачи для заданных значений шагов временной сетки. Улучшенная точность построенной схемы с оптимальным весом по сравнению с явной схемой и эффективность алгоритма поиска оптимального значения весового параметра продемонстрированы на примере тестовой задачи. A difference scheme with weights for a homogeneous spatially one-dimensional diffusion-convection equation is studied. An analysis of the approximation error for the difference scheme as a time step function is performed on the basis of the expansion of the solution and approximation error in a trigonometric basis. An algorithm is proposed to find the optimal weight value that ensures the minimum approximation error of the solution to an initial boundary value problem for the given values of the time grid steps. A better accuracy of the constructed scheme with the optimal weight compared to the explicit scheme as well as the efficiency of the algorithm for finding the optimal weight value is shown using a test problem.

COMBINED METHOD OF INTEGRAL TRANSFORMS FOR THE SPHERICALLY SYMMETRIC DROPLET HEATING PROBLEM

2021 ◽  
Vol 10 (9) ◽  
pp. 3141-3164
Author(s):  
Kwassi Anani
Keyword(s):  
Series Solution ◽  
Integral Transform ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Heat Diffusion ◽  
Spherically Symmetric ◽  
Time Step ◽  
Spherical Droplet ◽  
Heat Diffusion Equation ◽  
Gas Environment

In this paper, we analysed the spherically symmetric heat diffusion equation, which governs the temperature distribution inside a heated but non-evaporating droplet. The spherical droplet, with an initial uniform temperature, is assumed at rest in an unsteady gas environment. The classical Fourier sine integral transform (FSIT) and the unilateral Laplace integral transform (LIT) are successively used to solve the resulting initial-boundary value problem, first reduced in a dimensionless form. An explicit solution in the Laplace domain is obtained for the temperature inside the droplet. Then, depending on the time-varying temperature of the gas environment at the immediate vicinity of the droplet, an exact series solution and an approximate analytical solution in short time limits are derived for the droplet internal temperature. In the case of steady gas environment at constant temperature, the standard series solution obtained in the literature for the symmetrical problem of heating or cooling of a solid spherical body, is recovered. The results may be useful for time step analysis in droplets and sprays vaporization models.

scholarly journals Mathematical Modelling and Numerical Simulation of Diffusive Processes in Slow Changing Domains

2020 ◽  
Author(s):  
Dmytro V. Yevdokymov ◽  
Yuri L. Menshikov
Keyword(s):  
Heat Conduction ◽  
Time Scale ◽  
Numerical Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Numerical Approach ◽  
Elliptic Boundary Value ◽  
Initial Boundary Value

Nowadays, diffusion and heat conduction processes in slow changing domains attract great attention. Slow-phase transitions and growth of biological structures can be considered as examples of such processes. The main difficulty in numerical solutions of correspondent problems is connected with the presence of two time scales. The first one is time scale describing diffusion or heat conduction. The second time scale is connected with the mentioned slow domain evolution. If there is sufficient difference in order of the listed time scale, strong computational difficulties in application of time-stepping algorithms are observed. To overcome the mentioned difficulties, it is proposed to apply a small parameter method for obtaining a new mathematical model, in which the starting parabolic initial-boundary-value problem is replaced by a sequence of elliptic boundary-value problems. Application of the boundary element method for numerical solution of the obtained sequence of problems gives an opportunity to solve the whole considered problem in slow time with high accuracy specific to the mentioned algorithm. Besides that, questions about convergence of the obtained asymptotic expansion and correspondence between initial and obtained formulations of the problem are considered separately. The proposed numerical approach is illustrated by several examples of numerical calculations for relevant problems.

scholarly journals AN INTEGRO-DIFFERENTIAL CONSERVATION LAW ARISING IN A MODEL OF GRANULAR FLOW

2012 ◽  
Vol 09 (01) ◽  
pp. 105-131 ◽  
Author(s):  
DEBORA AMADORI ◽  
WEN SHEN
Keyword(s):  
Granular Flow ◽  
Conservation Law ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Standard Theory ◽  
Step Method ◽  
Fractional Step Method ◽  
Time Step ◽  
Global Existence Of Solutions

We study a scalar integro-differential conservation law which was recently derived by the authors as the slow erosion limit of a granular flow. Considering a set of more general erosion functions, we study the initial boundary value problem for which one cannot adapt the standard theory of conservation laws. We construct approximate solutions with a fractional step method, by recomputing the integral term at each time step. A prioriL∞bound and total variation estimates yield the convergence and global existence of solutions with bounded variation. Furthermore, we present a well-posedness analysis which establishes that these solutions are stable in the L1norm with respect to the initial data.

scholarly journals Parallel implementation of a meshfree method for calculating flows of ideal incompressible fluid

2017 ◽  
pp. 175-186
Author(s):  
В.Н. Говорухин
Keyword(s):  
Parallel Algorithm ◽  
Stream Function ◽  
Parallel Implementation ◽  
Flow Region ◽  
Meshfree Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Ideal Incompressible Fluid ◽  
Time Step ◽  
Initial Boundary Value

Предложен параллельный алгоритм для расчета двумерной динамики невязкой несжимаемой жидкости на вращающейся сфере. Основой алгоритма является бессеточный метод вихрей в ячейках для решения начально-краевой задачи для нестационарных уравнений движения идеальной жидкости в терминах абсолютной завихренности и функции тока. Метод базируется на аппроксимации функции тока отрезком ряда Фурье, приближении поля завихренности ее значениями в частицах и расчете траекторий частиц с использованием псевдосимплектического интегратора. Схема распараллеливания на каждом временном шаге включает в себя расщепление по подмножествам частиц и декомпозицию области течения. Представлено описание алгоритма для вычислительных систем с общей памятью. Эффективность метода и производительность параллельного алгоритма оценены экспериментально при различных параметрах расчета, показана хорошая масштабируемость алгоритма. A parallel algorithm for calculating the two-dimensional dynamics of inviscid incompressible fluids on a rotating sphere is proposed. The algorithm is based on the meshfree vortex-in-cell method for solving an initial boundary value problem for unsteady equations describing the motion of an ideal fluid in terms of the absolute vorticity and stream function. The method is based on the approximation of the stream function using the Fourier series. The vorticity field is defined by its values on a set of particles. The particle trajectories are calculated using a pseudo-symplectic integrator. At each time step, the parallelization involves the splitting into subsets of particles and the decomposition of the flow region. The efficiency of the parallel algorithm and its performance are evaluated experimentally for various parameters of the method. The numerical results show a good scalability of the algorithm.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

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