Try to Understand Boundary Conditions in FEM Using a Simple 1D Bar

In Finite Element Method (FEM), boundary conditions define how a structure interacts with its surroundings. They are essential because, without them, the FEM system cannot produce a unique physical solution.

Consider a simple 1D bar with 3 nodes and 2 elements:

Node 1        Node 2        Node 3
|-------------|-------------|

In this example:

• the left end is fixed,
• and a force is applied at the right end.

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Dirichlet Boundary Condition

At Node 1:

$$u_1 = 0$$

This means the displacement is fixed to zero. This is called a Dirichlet boundary condition because the displacement value is prescribed directly.

In the Python code:

fixed_dofs = [0]

This prevents the bar from moving freely.

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Neumann Boundary Condition

At Node 3:

$$F_3 = 1000\text{ N}$$

This specifies an external force acting on the bar. This is called a Neumann boundary condition.

In the code:

F = np.array([0, 0, 1000.0])

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FEM Equation

The FEM system is written as:

$$[K]\{u\} = \{F\}$$

where:

• $[K]$ is the stiffness matrix,
• $\{u\}$ is the displacement vector,
• $\{F\}$ is the load vector.

The local stiffness matrix for each element is:

The Python code assembles these into the global matrix and solves for the unknown displacements.

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Why Boundary Conditions Matter

Without the fixed support, the entire bar could move freely, making the stiffness matrix singular and the FEM problem unsolvable.

Boundary conditions:

• define supports and loads,
• remove rigid body motion,
• and ensure a meaningful solution.

In this simple example:

• the fixed support is a Dirichlet condition,
• the applied force is a Neumann condition.

Understanding these concepts is fundamental to learning FEM.