# 统计代写|主成分分析代写Principal Component Analysis代考|”Isoparametric Formulation of Four-Node Quadrilateral Element in Finite Element Analysis”

The text describes the development of the four-node quadrilateral element in finite element analysis using an isoparametric formulation suitable for both plane stress and plane strain cases. The quadrilateral element addresses some limitations of triangular and rectangular elements, offering better flexibility in approximating complex geometries while maintaining linear variation of strains within the element.

Here’s a summary of the key steps:

Geometric Mapping: The physical coordinates (x, y) of any point inside the quadrilateral element are determined by the interpolation functions Ni(r, s) and the nodal coordinates (xi, yi). The mapping between the parent (natural) and physical space is given by:

x = ΣNi(r, s)xi y = ΣNi(r, s)yi Displacement Interpolation: Displacements (u, v) at any point in the element are interpolated using the same Ni(r, s) functions: u(x, y) = ΣNi(r, s)ui v(x, y) = ΣNi(r, s)vi Strain Computation: Strain components {ε} are derived by taking derivatives of the displacements with respect to global coordinates. This requires transforming the derivatives with respect to natural coordinates to those with respect to global coordinates using the Jacobian matrix [J]:

{ε} = [G][P]{δ} where [G] is the geometric mapping matrix that depends on the determinant of the Jacobian, and [P] is the matrix of partial derivatives of the interpolation functions with respect to the natural coordinates.

Stiffness Matrix: The element stiffness matrix [k(e)] is computed by integrating the product of [BT][D][B] over the element area in the physical space, which involves transforming to the natural coordinates and using the determinant of the Jacobian:

[k(e)] = t ∫∫ [B]^T [D] [B] |J| dr ds Here, [D] is the material property matrix specific to plane stress or plane strain conditions, t is the element thickness, and the integration is carried out over the range of natural coordinates -1 to 1.

Numerical Integration: Exact integration of the stiffness matrix terms is generally intractable; hence, Gaussian quadrature is employed to approximate the integrals:

[k(e)] ≈ t ΣΣ Wi Wj [B(ri, sj)]^T [D] [B(ri, sj)] |J(ri, sj)| where Wi and Wj are the Gaussian quadrature weights, and (ri, sj) are the integration points.

An example is provided where the properties of steel (Young’s modulus E = 30(10)^6 psi and Poisson’s ratio ν = 0.3) are used for a plane stress problem, with the element’s geometric mapping and Jacobian explicitly calculated. The goal would be to compute the stiffness matrix using the appropriate interpolation functions, Jacobian determinants, and the provided material property matrix [D] at the relevant Gaussian integration points.

The calculation of the stiffness matrix proceeds by plugging the given mappings, derivatives, and the Jacobian determinant into the formula above and carrying out the numerical integration at the designated quadrature points.

### MATLAB代写

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