Continuum Problem Types

Description

TopOpt.jl provides standard continuum topology optimization problem domains for testing and comparing algorithms. This tutorial covers 2D and 3D problems including cantilever beams, MBB beams, L-beams, tie-beams, and INP file import.

Continuum problems model structures as continuous solid domains discretized into finite elements. Design variables are element densities (0 = void, 1 = solid).

Setup

using TopOpt

2D and 3D Point Load Cantilever

The point load cantilever is a standard benchmark: a beam fixed at one end with a point load at the free end.

E = 1.0     # Young's modulus in MPa
ν = 0.3     # Poisson's ratio
f = 1.0     # downward force in N (negative is upward)
nels = (160, 40)        # 2D: 160×40 elements
elsizes = (1.0, 1.0)    # element size in mm
order = :Linear         # shape function order
problem_2d = PointLoadCantilever(Val{order}, nels, elsizes, E, ν, f)

The Val{order} specifies shape function order:

• :Linear — bilinear (2D) or trilinear (3D) elements
• :Quadratic — biquadratic (2D) or triquadratic (3D) elements

For 3D problems:

nels_3d = (160, 40, 40)     # 3D: 160×40×40 elements
elsizes_3d = (1.0, 1.0, 2.0) # element size in mm
problem_3d = PointLoadCantilever(Val{order}, nels_3d, elsizes_3d, E, ν, f)

2D and 3D Half MBB Beam

The Half MBB (Messerschmitt–Bölkow–Blohm) beam is a simply-supported beam with a central point load. Only half the beam is modeled (symmetry):

nels = (60, 20)
elsizes = (1.0, 1.0)
order = :Quadratic
problem = HalfMBB(Val{order}, nels, elsizes, E, ν, f)

Boundary conditions:

• Left edge: roller support (vertical displacement fixed)
• Bottom right: pin support (both DOFs fixed)
• Top center: downward point load

The 3D variant uses 3-tuples for nels and elsizes.

2D L-Beam Problem

The L-beam shows stress concentration effects at the re-entrant corner:

order = :Quadratic
problem = LBeam(
    Val{order};
    length=100,
    height=100,
    upperslab=50,
    lowerslab=50,
    E=1.0,
    ν=0.3,
    force=1.0
)

Geometry:

        upperslab   
       ............
       .          .
       .          .
       .          . 
height .          .                     
       .          ......................
       .                               .
       .                               . lowerslab
       .                               .
       .................................
                    length

The load is applied at the midpoint of the “lowerslab” vertical edge.

2D Tie-Beam Problem

The tie-beam has distributed loading on specified elements:

order = :Quadratic
problem = TieBeam(Val{order})

• Fixed supports at both ends
• Distributed downward load on specified elements
• 2D only

Reading INP (Abaqus) Files

For complex geometries, import from .inp (Abaqus) files:

filename = joinpath(@__DIR__, "problem.inp")
problem = InpStiffness(filename)

Workflow:

1. Define geometry in CAD software (FreeCAD, SolidWorks)
2. Mesh and export as .inp format
3. Import into TopOpt.jl for optimization

The .inp file contains nodes, elements, materials, BCs, and loads.