Cantilever beam shock and vibration
1 Objective
Verify mode-based linear dynamics on a cantilever beam: a frequency extraction, a random response to a base-acceleration power spectral density, and a peak response to a base-acceleration shock spectrum.
Coreform IGA elements are Abaqus user elements. Abaqus/Standard will extract eigenmodes for them, but it does not post-process a random response or a response spectrum from those elements, so for an immersed model the frequency extraction is the last result the solver produces. Coreform computes the two remaining procedures from that modal basis instead.
That arrangement has two things to verify, and this study checks both.
- The implementation. On the body-fitted mesh Abaqus computes all three steps itself, so the Coreform random response and response spectrum can be compared against the solver’s own answers on an identical modal basis. Any difference here is implementation error alone.
- The discretization. The immersed Coreform IGA modal basis, post-processed by Coreform, is compared against the body-fitted Abaqus analysis end to end. Any difference here is the difference between two meshes.
2 Geometry
| Parameter | Value |
|---|---|
| Width \(b\) | \(0.1\ m\) |
| Height \(h\) | \(0.1\ m\) |
| Length \(L\) | \(1\ m\) |
The beam is the same one used by the static cantilever beam bending study, so the two chapters describe one physical beam under different analyses. The section is square, which makes every bending mode a degenerate pair and puts a torsional mode inside the analysis band.
3 Material
| Property | Value |
|---|---|
| Young’s modulus \(E\) | \(200 \times 10^{9}\ Pa\) |
| Poisson’s ratio \(\nu\) | 0 |
| Mass density \(\rho\) | \(8.00 \times 10^{3}\ kg/m^3^\) |
Poisson’s ratio is zero so that the three-dimensional solid reproduces the one-dimensional beam theory the accepted frequencies come from.
4 Loading & boundary conditions
The root face at \(z = 0\) is fully clamped. Both dynamic steps excite the structure through a primary base motion in the global \(y\) direction, the weak bending direction of the section.
| Quantity | Value |
|---|---|
| Analysis band | \(10\ Hz\) to \(2.00 \times 10^{3}\ Hz\) |
| Modes extracted | \(20\) |
| Modal damping \(\xi\) | 20.0 ^{-3} |
| Base acceleration PSD | 10.0 ^{-3} g²/Hz, flat |
| Frequency points per interval | \(20\) |
| Bias parameter | 3 |
| Modal summation | SRSS |
The random response is driven by a flat base-acceleration spectral density across the band, the shape a vibration qualification test specifies. The response spectrum is driven by a base-acceleration shock spectrum that rises to a plateau and rolls off.
| Magnitude (m/s²) | Frequency (Hz) |
|---|---|
| 20 | 10 |
| 200 | 100 |
| 200 | 600 |
| 60 | 2.00 ^{3} |
5 Mesh & discretization
Two discretizations of the same beam. The body-fitted case is a structured C3D8 mesh that Abaqus owns end to end. The immersed case is a Coreform IGA mesh on a rectilinear background grid, trimmed to the beam.
| Case | Discretization | Element size | Dynamics computed by |
|---|---|---|---|
abaqus_h0p1 |
Body-fitted Abaqus (C3D8) | \(50.0 \times 10^{-3}\ m\) | Abaqus/Standard |
immersed_h0p1 |
Immersed Coreform IGA | \(25.0 \times 10^{-3}\ m\) | Coreform, from the modal basis |
The immersed case uses a finer background cell than the body-fitted element size. A linear immersed basis needs four cells across the section to resolve bending; at two cells the fundamental frequency comes out several percent stiff.
6 Reference & accepted solutions
The fundamental bending frequency of a clamped-free Euler-Bernoulli beam is
\[ f_1 = \frac{\beta_1^2}{2 \pi L^2} \sqrt{\frac{E I}{\rho A}}, \qquad \beta_1 = 1.8751 \]
with \(I = b h^3 / 12\) the second moment of area and \(A = b h\) the cross-sectional area. A three-dimensional solid sits slightly above this value because beam theory omits shear flexibility.
The random response and response spectrum quantities have no closed form, so the body-fitted Abaqus analysis is the accepted reference for them. Because Abaqus also computes both procedures on the body-fitted modal basis, the Coreform implementations are additionally checked against the solver directly, which is a stronger statement than the mesh comparison.
7 Results
7.1 Implementation against Abaqus
On a given modal basis the Coreform procedures and Abaqus solve the same problem, so they must agree to solver precision. Abaqus produces displacement-based dynamics output even for user elements, so this comparison is available on both discretizations.
The reported quantity is the component the base motion excites, which is what this chapter’s results are about and the only component whose magnitude is set by the excitation rather than by cross-coupling.
| Procedure | Body-fitted | Immersed | Ratio |
|---|---|---|---|
| Random response | 212 ^{-9} | 213 ^{-9} | 1 |
| Response spectrum | 97.1 ^{-9} | 121 ^{-9} | 1.25 |
Both discretizations agree with the solver at the precision of the single-precision fields Abaqus writes, and they agree with each other: the immersed modal basis costs the implementation nothing. That parity is asserted by the study rather than left to the reader, because it is the claim that lets the immersed workflow stand in for a solver-computed one.
Judging the whole displacement field instead would let a small secondary component decide the number. The axial component here peaks near eight percent of the field peak, so points that clear a field-wide floor still sit in that component’s own round-off, and the worst of them grows with however many output points a mesh happens to carry. The immersed mesh has roughly eight times as many output points as the body-fitted one, which on a field-wide measure inflates its figure to around \(1.4 \times 10^{-6}\) while the excited component stays at \(210 \times 10^{-9}\) on both. The field-wide numbers are recorded in the downloadable data for transparency, but nothing is gated on them.
7.2 Eigenfrequencies
| Mode | Body-fitted (Hz) | Immersed (Hz) | Relative difference | \(|\Gamma_y|\) | Participating |
|---|---|---|---|---|---|
| \(1\) | 81.6 | 81.8 | 2.10 ^{-3} | 0.116 | yes |
| \(2\) | 81.6 | 81.8 | 2.25 ^{-3} | 6.99 | yes |
| \(3\) | 492 | 494 | 5.20 ^{-3} | 0.2 | yes |
| \(4\) | 492 | 494 | 5.50 ^{-3} | 3.89 | yes |
| \(5\) | 722 | 822 | 0.139 | 816 ^{-15} | no |
| \(6\) | 1.25 ^{3} | 1.25 ^{3} | 292 ^{-6} | 31.4 ^{-15} | no |
| \(7\) | 1.30 ^{3} | 1.32 ^{3} | 8.46 ^{-3} | 9.96 ^{-3} | no |
| \(8\) | 1.30 ^{3} | 1.32 ^{3} | 8.75 ^{-3} | 2.29 | yes |
Against the closed form, the body-fitted mesh gives \(81\.6\\\\\\\\ Hz\) and the immersed mesh \(81\.8\\\\\\\\ Hz\), where Euler-Bernoulli theory predicts \(80\.8\\\\\\\\ Hz\). Both sit above the beam-theory value, as a shear-flexible solid should.
7.3 Response
| Quantity | Body-fitted Abaqus | Immersed Coreform | Relative difference |
|---|---|---|---|
| RMS displacement, random response | \(340 \times 10^{-6}\ m\) | \(346 \times 10^{-6}\ m\) | 17.9 ^{-3} |
| Peak displacement, shock spectrum | \(965 \times 10^{-6}\ m\) | \(983 \times 10^{-6}\ m\) | 18.3 ^{-3} |
8 Discussion
The two halves of the arrangement behave differently, and separating them is the point of this study.
The implementation is exact to solver precision. Given the same modal basis, the Coreform random response and response spectrum reproduce Abaqus to a few parts in ten million, which is the precision at which Abaqus stores its fields. Nothing in the combination introduces error.
The discretizations differ by the amount two meshes differ. The participating eigenfrequencies agree to under one percent, and the RMS and peak responses to under two percent, on a comparison between a body-fitted linear hexahedral mesh and a trimmed immersed spline basis.
Not every mode needs to agree for the response to agree. The square section puts a torsional mode inside the analysis band, and a linear immersed basis represents torsional warping far less accurately than bending: that mode is stiff by an order of ten percent. It carries no participation in a transverse base motion, so it never reaches the response, and the study gates on the participating modes for that reason. The full spectrum is tabulated above so the departure is visible rather than hidden, and it is a useful reminder that agreement mode by mode is neither necessary nor sufficient for agreement in the response.
RMS values depend on the frequency sweep. The random response integrates a spectral density that is sharply peaked at lightly damped resonances, so the reported RMS is only as converged as the sweep beneath it. At the settings used here both codes integrate the same points and agree with each other, but neither is the converged integral. A production run should raise the number of points per interval until the RMS stops moving.
9 References
10 Appendix
10.1 Capabilities exercised
No qualifying capability evidence was generated for these run artifacts.
10.2 Supported Abaqus keywords
*Heading
*Preprint, echo=NO, model=NO, history=NO, contact=NO
*Part, name=<part>
*Node
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,
*End Part
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60., 2000., 0.02
*Material, name=<material>
*Density
8000.,
*Elastic
2e+11,0.
*Boundary
<node-set>, ENCASTRE
*Step, name=<step>, nlgeom=NO, perturbation
*Frequency, eigensolver=Lanczos, sim=NO, acoustic coupling=on, normalization=mass
20, , , , ,
*Restart, write, frequency=0
*Output, field
*Node Output, variable=PRESELECT
*Element Output, directions=YES
S,
*End Step
*Step, name=<step>, nlgeom=NO, perturbation
*Random Response
10., 2000., 20, 3.,
*Modal Damping, definition=FREQUENCY RANGE
10., 0.02
2000., 0.02
*Base Motion, type=ACCELERATION, dof=2, scale=1., load case=1
*Correlation, type=CORRELATED, psd=base-psd, complex=NO
1, 1.
*Output, field
*Node Output
RU, U
*Element Output, directions=YES
RMISES, RS
*Output, history, frequency=0
*End Step
*Step, name=<step>, nlgeom=NO, perturbation
*Response Spectrum, sum=SRSS
shock-spectrum,0., 1., 0., 1.
*Modal Damping, definition=FREQUENCY RANGE
10., 0.02
2000., 0.02
*Output, field
*Node Output
U,
*Element Output, directions=YES
S,
*Output, history, frequency=0
*End Step10.3 Download artifacts
Download all verification artifacts (ZIP)
10.3.1 Geometry CAD files
No separate CAD file is required for this problem; the geometry is fully defined in the Abaqus/CAE journal files.
10.3.2 Abaqus/CAE journal files
2 generated Abaqus/CAE journals are available.
Show 2 individual journals
10.4 Provenance
| Parameter | Value |
|---|---|
| Evidence type | Numerical verification |
| CTest result | Passed (Release) |
| Verification result | Passed — 4/4 enforced criteria passed |
| Test date | 30-Sep-2026 16:33:31 |
| Operating system | Ubuntu 22.04.4 LTS |
| Coreform IGA version | 2026.10+70894 |
| Source revision | 733d3350b486 |
| Abaqus version | Abaqus Unofficial Packaging Version |
| Parameter | Value |
|---|---|
| Evidence type | Numerical verification |
| CTest result | Passed (Release) |
| Verification result | Passed — 4/4 enforced criteria passed |
| Test date | 30-Sep-2026 16:33:52 |
| Operating system | Ubuntu 22.04.4 LTS |
| Coreform IGA version | 2026.10+70894 |
| Source revision | 733d3350b486 |
| Abaqus version | Abaqus Unofficial Packaging Version |