Implicit-dynamic vibration of a three-dimensional beam
1 Objective
Verify direct implicit-dynamic time integration for native Abaqus, body-fitted Coreform IGA, and immersed Coreform IGA discretizations. The six-case study exercises both the TRANSIENT_FIDELITY and MODERATE_DISSIPATION Abaqus application shortcuts [1].
2 Geometry
The model is a rod of length \(L=10\) aligned with the \(x\) axis and a \(2\times2\) square cross-section.
3 Material
The rod is linear elastic with Young’s modulus \(E=1\times10^6\), density \(\rho=1\times10^{-6}\), and Poisson’s ratio \(\nu=0\) in a dimensionally consistent unit system. The zero Poisson’s ratio makes the response effectively one-dimensional.
4 Loading & boundary conditions
The \(-x\) face is fixed, and the lateral faces are free. A pressure on the \(+x\) face rises smoothly from zero to its peak during one analytical fundamental-mode period, returns smoothly to zero during the next period, and then remains zero. The free-vibration comparison starts after the load has been removed.
5 Mesh & discretization
The native reference uses structured C3D8R elements with nominal size \(h=1\). Both Coreform IGA discretizations use quadratic, \(C^1\) splines with the same nominal size; one is body-fitted and the other is immersed.
All cases cover \(0\leq t\leq2\times10^{-3}\) with an initial increment of \(1\times10^{-6}\) and a maximum increment of \(2\times10^{-6}\). Axial displacement, velocity, and acceleration are recorded at every accepted increment and averaged over the free end.
6 Reference & accepted solutions
The fundamental axial frequency of a fixed-free rod is
\[ f_1=\frac{1}{4L}\sqrt{\frac{E}{\rho}}=25{,}000. \]
Each case must predict \(f_1\) within 2%. Each Coreform IGA history is also compared with the native Abaqus history using the same application shortcut.
| Quantity | Acceptance criterion |
|---|---|
| Fundamental-frequency relative error | \(<0.02\) |
| Normalized displacement-history RMS error | \(<0.01\) |
| Free-vibration phase RMS error | \(<0.2\ \text{rad}\) |
| Amplitude-decay-ratio difference | \(<0.03\) |
| Energy-decay-ratio difference | \(<0.05\) |
| Accepted-increment count ratio | \(\leq1.10\) |
| Samples per fundamental-mode period | \(\geq19.9\) |
7 Results
7.1 Fundamental frequency
| Application | Discretization | \(f_1\) | Relative error | Accepted increments |
|---|---|---|---|---|
TRANSIENT_FIDELITY |
Native Abaqus | 24745.6 | 1.02% | 1003 |
TRANSIENT_FIDELITY |
Body-fitted Coreform IGA | 24767.1 | 0.93% | 1003 |
TRANSIENT_FIDELITY |
Immersed Coreform IGA | 24766.9 | 0.93% | 1003 |
MODERATE_DISSIPATION |
Native Abaqus | 24681.0 | 1.28% | 1002 |
MODERATE_DISSIPATION |
Body-fitted Coreform IGA | 24702.4 | 1.19% | 1002 |
MODERATE_DISSIPATION |
Immersed Coreform IGA | 24702.2 | 1.19% | 1002 |
The maximum normalized displacement-history RMS error is \(2.21\times10^{-3}\), and the maximum free-vibration phase RMS error is \(0.155\ \text{rad}\). The maximum amplitude- and energy-decay-ratio differences are \(304\times10^{-6}\) and \(1.11\times10^{-3}\), respectively.
7.2 Representative displacement field
The final recorded TRANSIENT_FIDELITY immersed result is shown using the same CTest result database that supplies the verification evidence.
8 Discussion
Every spatial discretization and application shortcut predicts the analytical fundamental frequency within the enforced tolerance. The body-fitted and immersed histories track their matching native Abaqus references without using more accepted increments. The sampling criterion confirms that agreement is not obtained by taking substantially smaller time increments.
References
[1]
Abaqus analysis guide, 2026 FD03. Dassault Systèmes, 2026.
Appendix
8.1 Download artifacts
Download all verification artifacts (ZIP)
8.1.1 Geometry CAD files
8.1.2 Abaqus/CAE journal files
6 generated Abaqus/CAE journals are available.
8.2 Provenance
The records below identify the six simulation CTests and their cross-case study CTest.
NoteTest_abaqus_verification_vibrating_beam_3d_tf_native
| Parameter | Value |
|---|---|
| Evidence type | Numerical verification |
| CTest result | Passed (Release) |
| Verification result | Passed — 1/1 enforced criterion passed |
| Test date | 10-Sep-2026 16:03:33 |
| Operating system | Ubuntu 22.04.4 LTS |
| Coreform IGA version | 2026.9+69838 |
| Source revision | d88163e59ac0 |
| Abaqus version | Abaqus 6.26-4 |
NoteTest_abaqus_verification_vibrating_beam_3d_tf_bodyfit
| Parameter | Value |
|---|---|
| Evidence type | Numerical verification |
| CTest result | Passed (Release) |
| Verification result | Passed — 1/1 enforced criterion passed |
| Test date | 10-Sep-2026 15:55:40 |
| Operating system | Ubuntu 22.04.4 LTS |
| Coreform IGA version | 2026.9+69838 |
| Source revision | d88163e59ac0 |
| Abaqus version | Abaqus 6.26-4 |
NoteTest_abaqus_verification_vibrating_beam_3d_tf_immersed
| Parameter | Value |
|---|---|
| Evidence type | Numerical verification |
| CTest result | Passed (Release) |
| Verification result | Passed — 1/1 enforced criterion passed |
| Test date | 10-Sep-2026 15:56:41 |
| Operating system | Ubuntu 22.04.4 LTS |
| Coreform IGA version | 2026.9+69838 |
| Source revision | d88163e59ac0 |
| Abaqus version | Abaqus 6.26-4 |
NoteTest_abaqus_verification_vibrating_beam_3d_md_native
| Parameter | Value |
|---|---|
| Evidence type | Numerical verification |
| CTest result | Passed (Release) |
| Verification result | Passed — 1/1 enforced criterion passed |
| Test date | 10-Sep-2026 15:57:14 |
| Operating system | Ubuntu 22.04.4 LTS |
| Coreform IGA version | 2026.9+69838 |
| Source revision | d88163e59ac0 |
| Abaqus version | Abaqus 6.26-4 |
NoteTest_abaqus_verification_vibrating_beam_3d_md_bodyfit
| Parameter | Value |
|---|---|
| Evidence type | Numerical verification |
| CTest result | Passed (Release) |
| Verification result | Passed — 1/1 enforced criterion passed |
| Test date | 10-Sep-2026 15:57:11 |
| Operating system | Ubuntu 22.04.4 LTS |
| Coreform IGA version | 2026.9+69838 |
| Source revision | d88163e59ac0 |
| Abaqus version | Abaqus 6.26-4 |
NoteTest_abaqus_verification_vibrating_beam_3d_md_immersed
| Parameter | Value |
|---|---|
| Evidence type | Numerical verification |
| CTest result | Passed (Release) |
| Verification result | Passed — 1/1 enforced criterion passed |
| Test date | 10-Sep-2026 15:56:20 |
| Operating system | Ubuntu 22.04.4 LTS |
| Coreform IGA version | 2026.9+69838 |
| Source revision | d88163e59ac0 |
| Abaqus version | Abaqus 6.26-4 |
NoteTest_abaqus_verification_study_vibrating_beam_3d_core
| Parameter | Value |
|---|---|
| Evidence type | Numerical verification |
| CTest result | Passed (Release) |
| Verification result | Passed — 26/26 enforced criteria passed |
| Test date | 10-Sep-2026 16:03:45 |
| Operating system | Ubuntu 22.04.4 LTS |
| Coreform IGA version | 2026.9+69838 |
| Source revision | d88163e59ac0 |
| Abaqus version | Abaqus 6.26-4 |