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# 数学代写|有限元方法代写finite differences method代考|ENGR7961 Solution Process

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## 数学代写|有限元代写Finite Element Method代考|Solution Process

The significance of the information provided in the above input file is very similar to the previous case study. Therefore, this section will highlight the differences that are mainly used for the transient analysis.

The definition of amplitude curve is important here as it enables the load (or boundary condition) to be defined as a function of time here. In this case the load will follow the sinusoidal function defined in the amplitude curve block. The sinusoidal function is defined as a periodic function whereby the formula used is actually the Fourier series. The data lines in the amplitude curve block basically define the angular frequency and the other constants in the Fourier series.

The control card specifies that the analysis is a direct integration, transient analysis. In ABAQUS, Newmarks’s method (Section 3.7.2) together with the Hilber-Hughes-Taylor operator [1978] applied on the equilibrium equations is used as the implicit solver for direct integration analysis. The time increment is specified to be $0.1 \mathrm{~s}$, and the total time of the step is $1.0 \mathrm{~s}$. As mentioned in Chapter 3, implicit methods involve solving of the matrix equation at each individual increment in time, therefore the analysis can be rather computationally expensive. The algorithm used by ABAQUS is quite complex, involving the capabilities of having automatic deduction of the required time increments. Details are beyond the scope of this book.

## 数学代写|有限元代写Finite Element Method代考|Result and Discussion

Upon the analysis of the problem defined by the input file above, the displacement, velocity and acceleration components throughout each individual time increment can be obtained until the final time step specified. Therefore, we have what is known as the displacementtime history, the velocity-time history and the acceleration-time history, as shown in Figures $8.16,8.17$ and 8.18 , respectively. The plots show the displacement, velocity and acceleration histories of nodes 210 and 300 .

A three-dimensional (3D) solid element can be considered to be the most general of all solid finite elements because all the field variables are dependent of $x, y$ and $z$. An example of a 3D solid structure under loading is shown in Figure 9.1. As can be seen, the force vectors here can be in any arbitrary direction in space. A 3D solid can also have any arbitrary shape, material properties and boundary conditions in space. As such, there are altogether six possible stress components, three normal and three shear, that need to be taken into consideration. Typically, a 3D solid element can be a tetrahedron or hexahedron in shape with either flat or curved surfaces. Each node of the element will have three translational degrees of freedom. The element can thus deform in all three directions in space.

## MATLAB代写

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