NGSolve is a powerful finite element software for solving various types of partial differential equations (PDEs) and engineering problems, including beam deflection analysis. To perform a beam deflection analysis using NGSolve, you can follow these steps:

1. Install NGSolve and its dependencies by following the instructions in the NGSolve documentation (https://ngsolve.org/docu/latest/tutorial/install.html).
2. Define the geometry and mesh the beam using the appropriate meshing tools in NGSolve.
3. Define the material properties and boundary conditions for the beam.
4. Set up the finite element spaces and the linear system for the beam deflection analysis.
5. Assemble and solve the linear system using the appropriate solver.
6. Post-process and visualize the results, such as the deformed shape and stress/strain distribution of the beam.

In here, I described how to do simple 3D beam tensile test by NGSolve

import ngsolve as ngs
from ngsolve import grad

# Define the geometry and mesh the beam
geo = ngs.Geometry()
mesh = ngs.Mesh(geo.GenerateMesh(maxh=0.1))

# Define the finite element spaces and functions
fes = ngs.H1(mesh, order=1)
u, v = fes.TnT()

# Define the material properties
E = 2e5  # Young's modulus
nu = 0.3  # Poisson's ratio
mu = E / (2 * (1 + nu))
lamb = E * nu / ((1 + nu) * (1 - 2 * nu))

# Define the linear elasticity tensor
def sigma(v):
    return 2 * mu * sym(grad(v)) + lamb * tr(sym(grad(v))) * ngs.Id(v.dim)

# Define the boundary conditions
gfu = ngs.GridFunction(fes)
gfu.Set(0, definedon=mesh.Boundaries("left"))
gfu.Set(1, definedon=mesh.Boundaries("right"))

# Define the weak form and the linear system
a = ngs. BilinearForm(fes)
a += ngs.SymbolicBFI(sigma(u) * sigma(v))
f = ngs.LinearForm(fes)

# Assemble and solve the linear system
a.Assemble()
f.Assemble()
gfu.vec.data = a.mat.Inverse(fes.FreeDofs()) * f.vec

# Post-process and visualize the results
stress = sigma(gfu) - (1/3) * tr(sigma(gfu)) * ngs.Id(u.dim)
von_mises = ngs.sqrt(3/2 * sym(stress).Trace()**2)
von_mises.SetName("von_Mises_stress")
ngs.Draw(von_mises)
ngs.Draw(gfu, mesh, "displacement")
ngs.Draw(mesh)

This code defines the geometry and mesh of a 3D beam, sets up the finite element spaces and functions, defines the material properties and boundary conditions, and assembles and solves the linear system for the beam deflection analysis. It then post-processes and visualizes the results, such as the von Mises stress and the deformed shape of the beam.