Chapter 1 Preliminary Concepts

Last updated: 09 May 2026

1.5 Finite Element Analysis of Linear systems

1.5.1 Finite Element Approximation

Element level approximations (polynomial) of solutions

Linear polynomial approximation of 1D problem:

\[ u(x) = a_0 + a_1 x, \quad x \in [x_i, x_{i+1}] \]

which can be used to derive the expression for the approximate solution in terms of nodal values:

\[ u(x) = \frac{x_{i+1} - x}{L^e} u_i + \frac{x - x_i}{L^e} u_{i+1} \]

where \(L^e = x_{i+1} - x_i\) is the length of the element.

1.5.2 Finite Element Equations for a One-Dimensional Problem

The differential equation along with boundary conditions is called the boundary value problem (BVP).

In general, the stiffness matrix without imposing boundary conditions is singular.

1.5.3 Finite Element Equations for 3D Solid Element

The isoparametric mapping is not valid if the Jacobian is zero or negative anywhere in the element.

(interior point \(\boldsymbol{\zeta}\) \(\xrightarrow{mapping}\) exterior point \(\boldsymbol{x}\)) \(\Rightarrow\) negative Jacobian

the exterior point means outside of the physical element and vice versa.

(multiple points \(\boldsymbol{\zeta}\) \(\xrightarrow{mapping}\) single point \(\boldsymbol{x}\)) \(\Rightarrow\) zero Jacobian

The integration over the physical domain can be transformed to the reference domain using:

\[ \int_{\Omega^e} f(\boldsymbol{x}) d\Omega = \int_{\hat{\Omega}} f(\boldsymbol{x}(\boldsymbol{\zeta})) |J| d\hat{\Omega} \]

where \(|J|\) is the determinant of the Jacobian matrix of the transformation from the reference domain \(\hat{\Omega}\) to the physical domain \(\Omega^e\).

The assemble process can be denoted using the symbol \(\bigwedge(.)\)

In general, \(NG\)-points Gaussian integration method integrates \((2NG-1)\)-order plynomials exactly.

The computational cost of Gaussian integration is proportional to \(NG^2\) for 2D problems and \(NG^3\) for 3D problems.

1.5.4 Finite Element Equations for 2D Plane-Strain Quadrilateral Element

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