Finite

To evaluate thermal stress, it is necessary to know the temperature distribution inside the object, and the finite element method is an effective way to predict quantities such as internal temperature distribution and thermal stress. In this post, I intend to explore the one-dimensional heat transfer equation using the variational method. The temperature function and displacement function can be..

In this post, I aim to study the fundamental partial differential equation of one-dimensional heat transfer without convection. Through this, we can enhance our physical understanding of heat transfer phenomena. According to the law of energy conservation, the following relationship holds: Where, ${E_{{\text{in}}}}$ is represents the energy entering the volume, $U$ is the change in stored energy..

In the last post, I discussed the stress-strain of hexahedral elements, and in this post, I'll calculate the stiffness matrix of tetrahedral elements. Numbering each of the four nodes of the tetrahedron, the displacement at each node can be represented as follows. Since there are three degress of freedom per node, there are a total of 12 degrees of freedom. The element displacement functions $u$..

Consider an infinitesimal element of dimensions $dx$, $dy$ and $dz$ in a three-dimensional Cartesian coordinate system, subjected to normal and shear stresses. The normal stresses acting perpendicularly on the surfaces are denoted as ${\sigma _x}$, ${\sigma _y}$ and ${\sigma _z}$, while the shear stresses acting within the planes are represented as ${\tau _{xy}}$, ${\tau _{yz}}$ and ${\tau _{zx}..

Continuing from the previous post(38_High Order Shape Function), I intend to solve an example. The $B$ matrix and the $Jacobian$ could be calculated as follows. Substituting this into the stiffness matrix equation, For ease of integration, simplifying yields the following. Calculate it.. Therefore, the stiffness matrix can be expressed as follows.

While using higher-order shape functions has the advantage of more accurately approximating irregularly shaped curve boundaries, they are generally not used as frequently due to the increased computation time, with lower-order shape functions being the preferred choice. In this post, the method for calculating higher-order shape functions will be explained. As shown in the figure above, the aim ..

In this post, I'll continue solving the problem addressed in the previous post. Calculate the stress at Gauss points and represent the stress at the nodes. An example is provided to illustrate the method of using Gauss integration to determine the stress at the point $s=0$, $t=0$ of the element. Material properties are as follows. $E = 200{\text{GPa, v}} = 0.3,\,{u_1} = 0,\,{v_1} = 0,\,{u_2} = 0..

This post will solve an example of obtaining the stiffness matrix using the Gaussian method, as covered in the previous post. The stiffness matrix of the quadrilateral element is determined using the 4-point Gaussian quadrature rule. where $E=200 Gpa$ and $v=0.25$ Firstly, for a two-dimensional element, the stiffness matrix can be represented as follow. where $j$ is the $Jacobian$ matrix. Using ..

Gaussian quadrature, unlike Newton-Cotes quadrature, optimizes the position of sampling points instead of creating them at uniform intervals, allowing for more accurate integration. Therefore, Gaussian quadrature can perform more accurate integration with fewer sampling points. The generalization of the midpoint rule can be represented as follows. Where ${W_i}$ are the weights, and they are dete..

I'll study one of the numerical integration methods, the Newton-Cotes integration. The Newton-Cotes method approximates the integrand with a polynomial for integration, with the trapezoidal rule and Simpson's rule being typical examples. The Newton-Cotes method is not generally used as it does not yield results as accurate as the Gauss integration method. When there is the following definite int..