The finite element workflow: the BARC

This page builds a finite element model of plates in the app: the plates a structure is made of, meshed into shells, tied together where it is bolted, then solved. Further down, the same structure is built again from solid bricks, with its parts meshed apart and each joint made with Mate Surfaces over the nodes its bolts clamp. For a part too thick for plates, the solid workflow builds the Frame-Wing from solid shapes and meshes them into bricks.

The structure is the BARC (Box Assembly with Removable Component), a small bolted aluminum article the structural dynamics community shares as a common test piece. It is a square box tube whose top wall is slotted in two, with the Bench on top: two channels standing on the two halves and a flat bar bolted across their tops. Its solid model, test data and finite element models are shared on the SEM Dynamic Substructuring Focus Group wiki, https://wiki.sem.org/wiki/BARC. Every dimension below comes from that solid model, and the model was checked against the finite element models shared there.

Every step is done in the app. The whole model is also a demonstration module, visualdynamics.demo.barc, so you can check each step against it or skip ahead:

from visualdynamics.demo import barc

barc.project().save('barc.vdyn')    # the model this page builds, already built

Step	Where	How
Start a geometry	the project row's bar	+ New Geometry
Build it from plates	the geometry's Solids row	Add Plate, once per plate
Mesh the plates	the plates picked in the tree	Mesh Solids, Shells (quads)
Tie the bolted parts	the geometry's Quads	pick the elements under each washer, Tie to the part below
Give each part its material and thickness	the pencil on each element group's row	pick a material, type a thickness
Solve	the 3-D view's bar	Modes, then Solve Modes in the pane

1. The geometry, from plates

The simplest description of a plate model is its plates. Each plate is a rectangle at the part's mid-thickness, given as a center, two widths and a rotation. When the plates are meshed, each edge is divided evenly into the whole number of elements nearest the element size, so the elements come out close to square, and plates that meet along a line (the box's corners, a channel's web and flanges) share their nodes there. Each plate becomes an element group named for it, and plates given the same name are one element group: the box's five walls are the group box.

Choose the inch system in the unit menu, select the project's row and press + (New Geometry). This makes an empty geometry in inches, named Geometry. Expand it, select its Solids row and press Add Plate. A pane opens beside the 3-D view. Type each row of the table below (name, center, widths, rotation) and press Add. A plate lies in its own X-Y plane, so a rotation of 90° about X stands it in the X-Z plane and 90° about Y in the Y-Z plane. The pane stays open for the next row and draws the plate in the 3-D view as you type. You can also turn the preview about its center with the rings around it and slide the center along its axes with the arrows, which land on a tenth of an inch; the rows in the table do not need either.

Plate	Name	Center	Widths	Rotation
box, bottom	box	(0, −2.875, 1.5)	(5.75, 3)	(90, 0, 0)
box, right side	box	(2.875, 0, 1.5)	(3, 5.75)	(0, 90, 0)
box, left side	box	(−2.875, 0, 1.5)	(3, 5.75)	(0, 90, 0)
box, top left of the slot	box	(−1.5625, 2.875, 1.5)	(2.625, 3)	(90, 0, 0)
box, top right of it	box	(1.5625, 2.875, 1.5)	(2.625, 3)	(90, 0, 0)
right channel, foot	right channel	(1.96875, 3.0625, 1.5)	(0.9375, 1)	(90, 0, 0)
right channel, top	right channel	(1.96875, 4.9375, 1.5)	(0.9375, 1)	(90, 0, 0)
right channel, web	right channel	(2.4375, 4, 1.5)	(1, 1.875)	(0, 90, 0)
left channel, foot	left channel	(−2, 3.0625, 1.53125)	(1, 0.9375)	(90, 0, 0)
left channel, top	left channel	(−2, 4.9375, 1.53125)	(1, 0.9375)	(90, 0, 0)
left channel, web	left channel	(−2, 4, 1.0625)	(1, 1.875)	(0, 0, 0)
beam	beam	(0, 5.0625, 1.5)	(5, 1)	(90, 0, 0)

The two channels are turned a quarter turn from each other, as they are on the hardware. The lengths are the solid model's, in inches (the geometry stores them in SI). Up is +y, as in the solid model.

Then pick the twelve plates under Solids (click the first, Shift-click the last) and press Mesh Solids. With plates picked, the pane offers only shells. Leave Shells (quads), type 0.125 for the element size and press Mesh. The plates become four element groups under Quads: box, right channel, left channel and beam.

From a script, the same plates are project.add_solid_plate and project.mesh_solids, which is what each Add and the Mesh record:

import visualdynamics

project = visualdynamics.Project('BARC')
name = project.new_geometry(unit='in')
across, up = (90, 0, 0), (0, 90, 0)          # into X-Z, into Y-Z

plates = [
    ('box', (0, -2.875, 1.5), (5.75, 3), across),              # bottom
    ('box', (2.875, 0, 1.5), (3, 5.75), up),                   # right side
    ('box', (-2.875, 0, 1.5), (3, 5.75), up),                  # left side
    ('box', (-1.5625, 2.875, 1.5), (2.625, 3), across),        # top, left of the slot
    ('box', (1.5625, 2.875, 1.5), (2.625, 3), across),         # top, right of it
    ('right channel', (1.96875, 3.0625, 1.5), (0.9375, 1), across),     # its foot
    ('right channel', (1.96875, 4.9375, 1.5), (0.9375, 1), across),     # its top
    ('right channel', (2.4375, 4, 1.5), (1, 1.875), up),                # its web
    ('left channel', (-2, 3.0625, 1.53125), (1, 0.9375), across),
    ('left channel', (-2, 4.9375, 1.53125), (1, 0.9375), across),
    ('left channel', (-2, 4, 1.0625), (1, 1.875), (0, 0, 0)),
    ('beam', (0, 5.0625, 1.5), (5, 1), across),
]
ids = [project.add_solid_plate(name, center, widths, turn, plate, unit='in')
       for plate, center, widths, turn in plates]
project.mesh_solids(name, ids, 0.125, unit='in', elements='quads')

The demonstration module builds the same mesh directly from mesh.plane, each plate a corner, two edges and the element size, and mesh.assemble.

2. The bolted joints

Where the parts are bolted, their mid-surfaces do not meet. A channel's foot sits 0.1875 in above the box wall's mid-surface, so the two share no nodes and something has to join them. That is a rigid, massless link: a two-node line whose second node moves exactly as the first does, carried by its rotation, with no mass added. An element group of two-node lines given the material rigid (massless) is a set of these links.

How the links are laid out is the most important modeling decision in this model, and it was settled by comparing against the models on the wiki. A single link at each bolt comes out soft, and softer with every mesh refinement, because a point tie on a plate is a local singularity and not a joint. Tying each foot over its whole area comes out stiff. Tied over the washer's area, the model agrees with the wiki's models and converges. The patch is made of the elements the washer covers, rather than every node within its radius, because elements are what you pick in the view: whole elements that can be counted.

So under each bolt, every node of the elements it clamps is tied to the nearest node of the part below. There are sixteen bolts: four through each channel's foot into the box, and above them four through the beam into each channel's top. Under a foot the patch is the washer's area, which is the element the bolt passes through and the eight around it. Through the beam the patch is only the element the bolt passes through. Four bolts at each end of the beam clamp as much as one bolt over its washer would, and wider patches on all four bolts make the beam's joint stiffer than the models on the wiki. At the 0.125 in mesh the patches are:

Bolts	Where, (x, z)	The elements under each	Tie to
left foot	(−2.1875, 1.3125), (−2.1875, 1.6875), (−1.8125, 1.3125), (−1.8125, 1.6875)	the element the bolt passes through and the eight around it	box
right foot	(1.8125, 1.3125), (1.8125, 1.6875), (2.1875, 1.3125), (2.1875, 1.6875)	the element the bolt passes through and the eight around it	box
beam, left end	the left foot's four, (x, z)	the element the bolt passes through	left channel
beam, right end	the right foot's four, (x, z)	the element the bolt passes through	right channel

A foot's mesh is 8 × 8, and its four patches are the 6 × 6 inside its outer ring of elements, split into four 3 × 3 squares. On the beam, the bolts are directly above the feet's bolts, each in the middle of one element.

In the app, expand the geometry and click the pencil on Quads. For each bolt, click one element of its patch in the 3-D view and Shift-click the rest (Cmd-click also works on a Mac, Ctrl-click on Windows and Linux). Then press Tie on the bar and pick the part below from its menu: box for a foot, left channel or right channel for the beam. Every node of the patch is linked to the nearest node of that part. The links go into an element group of rigid, massless links, ties, which the first Tie makes and every later Tie adds to. Sixteen Ties make the 160 links. For a joint where the nearest node of a whole part might be the wrong one, A second selection… in the same menu ties the patch to a second patch that you pick next.

The same project with every step below already done, modes solved, is on the website's downloads page, and is barc.project(solved=True).

3. Materials, thicknesses, solve

Expand the geometry in the tree (yours, or BARC), then Quads, select the element groups' rows (Shift-click to take them all) and click the pencil on one of them. The Element Groups table opens with a column for each group selected and, below Name and Elements, a row for each property in the display units.

Element group	Material	Thickness
box	6061-T6	0.25 in
right channel, left channel, beam	6061-T6	0.125 in
ties (bolts in the demo's BARC)	rigid (massless), set by Tie	—

Picking a material from the drop-down fills in its modulus, Poisson's ratio and density; you can type over any of them.

Once an element group has a material, the status bar gives its weight when you select it, the total of several when you select several, and the whole geometry's weight when you select the geometry. It is the mass the solve uses, shown in lbf when the display units are imperial and in kg when they are SI. Groups that have no material yet are left out, and the status bar says how many of them there are.

Select the geometry and turn on Modes on the 3-D view's bar. The Modes pane opens on Finite element, with the highest frequency to solve for (2000 Hz) and the damping to give every mode (2 % of critical) set already; press Solve Modes. The model has 5,470 nodes, 5,136 plates and 160 rigid links, and a mass of 0.820 kg; the solid model's volumes give 0.816 kg, the difference being that the plates have no bolt holes. It solves in a few seconds, and its modes go into the geometry's group: six rigid-body modes at 0 Hz, then the elastic modes. Plates typed from the table above and meshed at 0.125 in give the demo's nodes and elements, and the patches above are the demo's element for element, so a geometry built by hand solves to the same modes.

What was checked

The model's modes were checked against the finite element models shared on the wiki, and the joints described above are the ones that agreed with them. Those checks held the beam by one bolt at each end. With the four bolts used here, each over the element it passes through, the first ten elastic modes move by 0.2 % at most and keep their shapes. The model is not tuned toward any one of the shared models, since a model tuned to agree is no longer a check. To compare against a measurement of a particular unit, measure that unit and model what was measured: hardware differs from its drawing, the box's wall thickness most of all.

From a script:

from visualdynamics.demo import barc

model = barc.build()
shapes = model.eigensolution(maximum_frequency=2000)
print(shapes.frequency[6:16])                 # the first ten elastic modes

The BARC from bricks, bolted with Mate Surfaces

The same structure is also built from solid bricks, the same way the Frame-Wing is: each part made from solid shapes, blocks cut and joined, then meshed into eight-node bricks. This is the higher-fidelity model, and the removable component (the two channels and the beam) comes with it as a model of its own.

from visualdynamics.demo import barc

barc.solid_project(solved=True).save('barc-bricks.vdyn')  # both, solved

Bricks touch where the parts touch: a channel's foot sits flat on the box's top wall, and the beam sits flat on the channels' tops. Each part is meshed with its own nodes, which lie on the next part's nodes where the two touch but are joined to none of them. Each joint is then made with Mate Surfaces: it finds where the two parts touch, you keep the nodes the bolts clamp, and the joint holds them to the part below, rigidly or with a stiffness. Everywhere else the parts touch, the nodes stay free to separate, as the parts are between their bolts. A joint is kept under the geometry's Mated Surfaces row, where it can be opened again to change its nodes or its stiffness.

1. The parts, as solids

Choose the inch system, start a geometry (+ on the project's row), and select its Solids row. Add each block below with Add Solid Block: the pane takes the block's center, its three widths and a name, and draws it as you type. Each part is made of as few blocks as it can be. The box is one block less its inside and the slot in its top, a channel is one block less the inside of its C, and the beam is one block. These are the parts' true outlines, not their mid-surfaces: the box is 6 in square outside, a foot's underside is the box's top face at y = 3, and the beam's underside is the channels' top faces at y = 5.

Block	Center	Widths	Name
the box	(0, 0, 1.5)	6 × 6 × 3	box
its inside	(0, 0, 1.5)	5.5 × 5.5 × 4
the slot in its top	(0, 2.8125, 1.5)	0.5 × 0.625 × 4
right channel	(2, 4, 1.5)	1 × 2 × 1	right channel
its inside	(1.8125, 4, 1.5)	1.125 × 1.75 × 1.5
left channel	(−2, 4, 1.5)	1 × 2 × 1	left channel
its inside	(−2, 4, 1.6875)	1.5 × 1.75 × 1.125
beam	(0, 5.0625, 1.5)	5 × 0.125 × 1	beam

Each inside block reaches past its part's open faces so that it cuts all the way through. The box's inside cuts through its front and back. The right channel's inside cuts through its inboard side, where it has no web. The left channel is turned a quarter turn with its web at the back, so its inside cuts through its front. Pick the box's three blocks and press Subtract: the inside and the slot are taken from the box, the first of the three in the tree, and the part keeps its name, box. Pick each channel's two blocks and press Subtract to make one part each, right channel and left channel. The pencil on any block opens it again so you can change it, and its part changes with it.

Then mesh them. Pick the four parts, press Mesh Solids, type 0.125 for the element size, tick Keep the parts apart from each other, and press Mesh. The four parts are meshed on one grid, cut at every part's faces, so where two parts touch their nodes lie on each other's at any element size. Each part keeps its own nodes there (a channel's foot shares none of them with the box's top face), ready for the joints. Meshing the parts one at a time puts each on its own grid, and then the nodes do not line up: at a quarter inch, a channel's foot and the box's top have no nodes in the same place.

2. The joints

The bolts are the same sixteen as in the plate model, at the same (x, z): four through each foot into the box and four through the beam into each channel's top. Each foot is one joint and each end of the beam another, four joints in all. A joint holds the nodes its bolts clamp, by the same rule the plate model's patches follow. Under a foot that is each washer's area: the corners of the brick the bolt passes through and of the eight around it, 4 × 4 nodes. Through the beam it is only the four corners of the brick the bolt passes through. Holding only four nodes at a foot reads up to 10 % soft against the models on the wiki and loses the ninth mode's shape.

In the tree, select the box's and the right channel's element groups under Hexes (click one, then Cmd-click the other; Ctrl-click on Windows and Linux) and press Mate Surfaces on the bar. A pane opens beside the view, both parts are drawn see-through, and the patch where the foot rests on the box is listed and marked: 81 nodes, 9 × 9, on each side. Pick nodes of is on box, the patch's first side, so only the box's nodes are drawn. A foot's four bolts, 4 × 4 nodes each, share their middle rows and together make one 7 × 7 square, which is every node except the foot's outer ring. Click the 32 marks around the foot's edge to take them off; each lights up as the cursor reaches it. Leave Normal and Shear on Rigid, type right foot bolts as the name, and press Keep. Do the same for the box and the left channel, named left foot bolts.

Then select the right channel and the beam and press Mate Surfaces. The patch is the 81 nodes where the beam lies on the channel's top. Press None to take every node off, click the four corners of the brick each of the four bolts passes through (16 marks) to put them back, name it right top bolts, and press Keep. Repeat for the left channel and the beam. The four joints tie the four parts into one structure at 130 nodes.

A joint's stiffness is set the same way. Open a joint with the pencil on its row, set Normal or Shear to Stiffness, type a value per node in the display units, and press Save. This is how a bolted joint is tuned until the model's modes match the measured ones.

3. The bolts' mass, the material, solve

The bolts are point masses, as in the shared model: one point element at each bolt's head, on the top face of a foot or of the beam above the bolt, in an element group given the mass in the Element Groups table's Mass column. Click the pencil on the geometry's Points row and switch on add mode (+). With New element group chosen beside it, click a corner of the brick each foot bolt passes through on the foot's top face; then choose New element group again for the beam's eight bolts. A #8 screw at a foot is 0.00744 lb (1.927e-5 lbf·s²/in in the inch system). The beam's bolts share the shared model's single 1/4 in screw per channel, 0.0125 lb, so each carries 0.003125 lb (8.094e-6 lbf·s²/in). Leaving the bolt masses out raises the first ten modes by 1 to 7 %.

Give the box, the channels and the beam 6061-T6 in the Element Groups table (bricks take only a material) and press Solve Modes in the Modes pane. At an eighth of an inch the assembly has 15,429 nodes and 9,440 bricks and solves in a few seconds. Its first ten elastic modes are within 3.4 % of a brick model that joined each foot over its whole footprint; that model was checked against the wiki's model and test, and its frequencies were 1 to 4 % above the measured modes. The mode shapes stay in the same order (MAC 0.97 and up). The removable component, built by the same steps without the box, is within 1.2 % of its checked model.