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The process is repeated as shown in Figure 5.1 until the change in the solution values becomes very small, closer to the exact solution.įigure 5.1: Calculation results using Jacobi method The second iterated solution is x = 1.5 and Y = 2.
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When this solution is substituted back into system (2) : Here, zero is entered initial values for x and y.įor the first iteration, the solution is x = 1 and y = 1. To solve the system (2) by using the iterative method, give x and y initial values. For a better understanding of how the solution is obtained iteratively, the system (1) is transformed as shown below: The exact solution to this system of equations is x = 3 and y = 4. A simultaneous system of linear equations with two unknowns is solved using the Jacobi iterative method. In a thermo-fluid analysis, the iterative method is generally used. The first method is called the direct method, and the second method is called the iterative method. One of two different solver methods is commonly used. The group of equations can be expressed in matrix form and the solver used to determine the solutions to the equations is called the matrix solver. Therefore, if the model consists of one million elements, one million simultaneous linear equations with one million unknowns per variable would have to be solved. This relationship equation is created for each element. For the flow equation, the constant is the external force acting on Element3 while the rate of heating is the constant for the thermal equation. Therefore, for the three-dimensional example, the total number of the neighboring elements is six.
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For a three-dimensional example, the neighboring elements of Element3 would be not only the aforementioned four elements but also two more elements at the front and back of Element3. Element3 contacts with four total elements on the left, right, top, and bottom. The following expression is satisfied.įigure 5.20: Positional relationship of an element with neighboring elements, and the relationship equation For example, consider Element3 shown in Figure 5.20. ISBN 978-8-6.As seen in the previous section, the parameter values of each element are determined as a function of the values in the neighboring elements.