Determination of the Steady State Fiber Orientation Tensor States in Homogeneous Flows with Newton–Raphson Iteration Using Exact Jacobians
Abstract
1. Introduction
2. Materials and Methods
2.1. Representation of Fiber Orientation State
2.2. Determination of Steady State Orientations
2.3. Fiber Orientation Modelling in the Dilute Regime
2.4. Fiber Orientation Modelling in Semi-Dilute and Concentrated Regime
2.4.1. The Folgar-Tucker Diffusion Model
2.4.2. Strain Reduction Factor (SRF) Model
2.4.3. Reduced Strain Closure (RSC) Model
2.4.4. Retarding Principal Rate (RPR) Model
2.4.5. Anisotropic Rotary Diffusion (ARD) Models
2.4.6. Nematic Potential (NEM) Model
2.5. Closure Approximations and Their Explicit Derivatives
2.5.1. Quadratic Closure Approximation
2.5.2. Linear Closure Approximation
2.5.3. Hybrid Closure Approximation
2.5.4. Hinch and Leal Closure Approximation
WF | ISO | ||||||||
LIN | |||||||||
QDR | |||||||||
SF | SF2 | ||||||||
HL | HL1 | ||||||||
HL2 |
2.5.5. Eigenvalue-Based Fitted (EBF) Closure Approximations
2.5.6. Invariant-Based Fitted (IBF) Closure Approximations
2.6. Error Estimate
3. Results
3.1. Validation of Exact Jacobians Based on Finite Difference Approximation
3.2. Comparison of the NR Method and the Dormand-Prince Runge–Kutta (RK45) Method
- L1: Simple shear flow in the 1–2 plane, .
- L2: Balanced shear/planar-elongation flow, simple shear in 1-2 plane superimposed on planar elongation in 1–2 plane. given that .
3.3. Performance of NR Method Using Various Closure Approximations
3.4. Homogenous Flow Considerations
- SS: Simple shear flow in the 2–3 plane, i.e., .
- SUA: Two stretching/shearing flows, including simple shear in 2–3 plane superimposed with uniaxial elongation in 3-direction, i.e., . Two cases are considered, including balanced shear/stretch with and dominant stretch with .
- UA: Uniaxial elongation flow in the 3—direction, i.e., .
- BA: Biaxial elongation (BA) flow in the 2–3 plane, i.e.,
- PST: Two shear/planar-elongation flows, including simple shear in 2–3 plane superimposed on planar elongation in 1–3 plane, . Two cases are considered, including balanced shear-planar elongation with and dominant planar elongation with .
- SBA: Balanced shear/bi-axial elongation flow, including simple shear in 2–3 plane superimposed on biaxial elongation, i.e., . Two cases are considered for the shear to extension rate, and .
4. Conclusions
- The computed exact Jacobians aligned closely with finite difference approximations for all fiber orientation tensor models and 4th order fiber orientation tensor closure approximations considered, and the degree of alignment depended on the perturbation size, finite difference type and finite difference order of approximation.
- The steady state orientation predictions from the NR solutions matched closely with the results predicted by the RK45 method for all fiber orientation tensor models, 4th order fiber orientation tensor closure approximations and homogenous flow types considered in this paper. The NR method was observed to be comparatively faster than the RK45 method for computing steady state values of the second order orientation tensor in all cases.
- The NR convergence and stability behavior depended on the type and complexity of the fiber orientation tensor model, the initial guess of the orientation tensor and the material properties of the underlying composite material. A good initial guess resulted in faster and more stable convergence to physical steady state solutions.
- The NR convergence behavior also depended on the flow type and extension-to-shear rate ratio. Higher shear rate dominance and flow symmetry increased the NR convergence rate and vice versa.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
Appendix A.1. Eigenvalues and Eigenvector Derivatives
- ‘Mass’ Normalization
- 2.
- Preassigning an mth Component of
- 3.
- Predefining the Euclidean norm of
Appendix B
Appendix B.1. Optimal Fitted Closure Approximation Constants/Coefficients
Appendix B.1.1. Eigenvalue-Based Optimal Fitting Closure (EBOF) Approximation
- Linear Orthotropic Fitted Closure
- 2.
- Quadratic Orthotropic Fitted Closures
- 3.
- Cubic Orthotropic Fitted Closures
- Non-rational Fitted Closure
- Rational Fitted Closure
- 4.
- Quartic Orthotropic Fitted Closures
Appendix B.1.2. Invariant-Based Optimal Fitting Closure (IBOF) Approximation
1 | 2 | 3 | |
---|---|---|---|
0 | 2.49409081657860 × 101 | −4.97217790110754 × 10−1 | 2.34146291570999 × 101 |
1 | −4.35101153160329 × 102 | 2.34980797511405 × 101 | −4.12048043372534 × 102 |
2 | 7.03443657916476 × 103 | 1.53965820593506 × 102 | 5.73259594331015 × 103 |
3 | 3.72389335663877 × 103 | −3.91044251397838 × 102 | 3.19553200392089 × 103 |
4 | −1.33931929894245 × 105 | −2.13755248785646 × 103 | −6.05006113515592 × 104 |
5 | 8.23995187366106 × 105 | 1.52772950743819 × 105 | −4.85212803064813 × 104 |
6 | −1.59392396237307 × 104 | 2.96004865275814 × 103 | −1.10656935176569 × 104 |
7 | 8.80683515327916 × 105 | −4.00138947092812 × 103 | −4.77173740017567 × 104 |
8 | −9.91630690741981 × 106 | −1.85949305922308 × 106 | 5.99066486689836 × 106 |
9 | 8.00970026849796 × 106 | 2.47717810054366 × 106 | −4.60543580680696 × 107 |
10 | 3.22219416256417 × 104 | −1.04092072189767 × 104 | 1.28967058686204 × 104 |
11 | −2.37010458689252 × 106 | 1.01013983339062 × 105 | 2.03042960322874 × 106 |
12 | 3.79010599355267 × 107 | 7.32341494213578 × 106 | −5.56606156734835 × 107 |
13 | −3.37010820273821 × 107 | −1.47919027644202 × 107 | 5.67424911007837 × 108 |
14 | −2.57258805870567 × 108 | −6.35149929624336 × 107 | −1.52752854956514 × 109 |
15 | −2.32153488525298 × 104 | 1.38088690964946 × 104 | 4.66767581292985 × 103 |
16 | 2.14419090344474 × 106 | −2.47435106210237 × 105 | −4.99321746092534 × 106 |
17 | −4.49275591851490 × 107 | −9.02980378929272 × 106 | 1.32124828143333 × 108 |
18 | −2.13133920223355 × 107 | 7.24969796807399 × 106 | −1.62359994620983 × 109 |
19 | 1.57076702372204 × 109 | 4.87093452892595 × 108 | 7.92526849882218 × 109 |
20 | −3.95769398304473 × 109 | −1.60162178614234 × 109 | −1.28050778279459 × 1010 |
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ARD Parameters | ||||||
---|---|---|---|---|---|---|
- | ||||||
, | ||||||
- | ||||||
HYB1 | HYB2 | ISO | LIN | QDR | SF2 | HL1 | HL2 | |
---|---|---|---|---|---|---|---|---|
AT | 0.6436 | 0.9385 | 0.2220 | 0.4188 | 0.2691 | 2.0949 | 0.9618 | 4.3940 |
PT | 0.8088 | 0.7549 | 0.5837 | 0.5003 | 0.4244 | 1.6776 | 0.8241 | 3.4809 |
iARD | 0.5737 | 1.2712 | 0.3444 | 0.6336 | 0.5728 | 0.5100 | 0.8774 | 1.6148 |
pARD | 0.7169 | 0.5475 | 0.2722 | 0.2438 | 0.4155 | 1.4805 | 0.9818 | 3.6185 |
WPT | 0.8563 | 1.0386 | 0.3773 | 0.2525 | 0.2926 | 1.4543 | 0.9632 | 3.5284 |
Dz | 0.5899 | 0.8248 | 0.2373 | 0.5484 | 0.3233 | 0.7137 | 1.0594 | 2.7732 |
NEM | 0.6490 | 0.9306 | 0.4012 | 0.4314 | 0.1612 | 2.1012 | 0.9846 | 4.4062 |
pARD-RSC | 1.0030 | 1.3343 | 1.3062 | 1.0506 | 1.3699 | 0.5378 | 1.3441 | 1.6482 |
iARD-RPR | 0.5645 | 0.6900 | 0.3478 | 0.3687 | 0.5222 | 1.0731 | 1.0324 | 2.0512 |
NAT1 | NAT2 | ORS | ORT | ORW | ORW3 | |
---|---|---|---|---|---|---|
AT | 0.3557 | 0.5812 | 0.4573 | 0.3351 | 0.5994 | 0.6496 |
PT | 0.3076 | 0.3946 | 0.3327 | 0.2517 | 0.5693 | 0.5284 |
iARD | 0.2512 | 0.2663 | 0.2129 | 0.1934 | 0.4762 | 0.4213 |
pARD | 0.2975 | 0.3956 | 0.3216 | 0.2479 | 0.5354 | 0.5229 |
WPT | 0.3040 | 0.4068 | 0.3430 | 0.2622 | 0.5720 | 0.5612 |
Dz | 0.3532 | 0.3276 | 0.2805 | 0.2920 | 0.6704 | 0.6010 |
NEM | 0.3649 | 0.5869 | 0.4615 | 0.3388 | 0.6062 | 0.6520 |
pARD-RSC | 0.2529 | 0.2800 | 0.2074 | 0.1802 | 0.4772 | 0.4155 |
iARD-RPR | 0.1964 | 0.1709 | 0.1601 | 0.1404 | 0.3687 | 0.2820 |
IBOF | WTZ | LAR32 | VST | FFLAR4 | LAR4 | |
---|---|---|---|---|---|---|
AT | 6.1748 | 4.3147 | 5.0800 | 3.1567 | 4.3188 | 4.3101 |
PT | 5.6437 | 4.0286 | 4.7162 | 2.9443 | 4.0435 | 3.9967 |
iARD | 4.4942 | 3.0776 | 3.6782 | 2.2662 | 3.0115 | 3.0665 |
pARD | 5.5458 | 3.8741 | 4.5415 | 2.8548 | 3.8500 | 3.8818 |
WPT | 5.6412 | 4.0391 | 4.7234 | 2.9350 | 4.0486 | 3.9851 |
Dz | 6.4226 | 4.8010 | 5.5454 | 3.3924 | 4.8016 | 4.6691 |
NEM | 6.1978 | 4.3254 | 5.0936 | 3.1653 | 4.3271 | 4.3182 |
pARD-RSC | 4.1821 | 2.9401 | 3.4878 | 2.1358 | 2.9236 | 2.9070 |
iARD-RPR | 3.4882 | 2.4509 | 2.9082 | 1.7359 | 2.3590 | 2.4110 |
FT | 0.0015 | 0.0003 | 0.0000 |
PT | 0.0046 | 0.0021 | 0.0031 |
iARD | 0.0009 | 0.0006 | 0.0000 |
pARD | 0.0006 | 0.0003 | 0.0032 |
WPT | 0.0043 | 0.0020 | 0.0030 |
Dz | 0.0021 | 0.0011 | 0.0021 |
MRD | 0.0015 | 0.0008 | 0.0046 |
FT | PT | iARD | pARD | WPT | Dz | MRD | |
---|---|---|---|---|---|---|---|
NR | 0.7210 | 0.4075 | 0.5916 | 0.3784 | 0.3038 | 0.4615 | 0.7908 |
RK45 | 3.5829 | 3.9775 | 3.0067 | 2.6441 | 2.7791 | 3.4486 | 6.7519 |
L1 | L2 | |||||
---|---|---|---|---|---|---|
RSC | 0.0374 | 0.0058 | 0.0000 | 0.0178 | 0.0122 | 0.0017 |
FT | 0.0036 | 0.0005 | 0.0000 | 0.0015 | 0.0011 | 0.0000 |
SRF | 0.0374 | 0.0055 | 0.0024 | 0.0158 | 0.0109 | 0.0050 |
RPR | 0.0374 | 0.0058 | 0.0000 | 0.0178 | 0.0122 | 0.0017 |
M1 | M2 | M3 | |
---|---|---|---|
1/30 | 1/30 | 1/20 | |
M1 | M2 | M3 | |
---|---|---|---|
0.0165 | 0.0630 | 0.0060 | |
0.9990 | 1.0100 | 0.9000 | |
0.9880 | 0.9650 | 0.9000 | |
0.9650 | 0.9650 | 0.9500 | |
0.0000 | 0.0000 | 0.0000 |
iARD-RPR | pARD-RPR | iARD-RSC | |||||||
---|---|---|---|---|---|---|---|---|---|
M1 | M2 | M3 | M1 | M2 | M3 | M1 | M2 | M3 | |
0.1039 | 0.0757 | 0.0876 | 0.0878 | 0.0278 | 0.0971 | 0.1107 | 0.0395 | 0.3231 | |
0.0550 | 0.0255 | 0.0219 | 0.0444 | 0.0131 | 0.0245 | 0.0553 | 0.0202 | 0.0678 | |
0.1286 | 0.1703 | 0.0401 | 0.0656 | 0.0270 | 0.0415 | 0.0872 | 0.0450 | 0.0235 |
(a) | (b) | ||||||
---|---|---|---|---|---|---|---|
HYB1 | 0.0042 | 0.0016 | 0.0000 | IBOF | 0.0068 | 0.0000 | 0.0000 |
HYB2 | 0.0000 | 0.0000 | 0.0000 | ORS | 0.0000 | 0.0000 | 0.0000 |
ISO | 0.0000 | 0.0000 | 0.0000 | ORT | 0.0061 | 0.0013 | 0.0000 |
LIN | 0.0000 | 0.0000 | 0.0000 | NAT1 | 0.0000 | 0.0000 | 0.0000 |
QDR | 0.0000 | 0.0000 | 0.0000 | ORW | 0.0067 | 0.0000 | 0.0000 |
SF2 | 0.0055 | 0.0000 | 0.0000 | NAT2 | 0.0072 | 0.0000 | 0.0000 |
HL1 | 0.0042 | 0.0014 | 0.0000 | WTZ | 0.0060 | 0.0000 | 0.0000 |
HL2 | 0.0038 | 0.0000 | 0.0000 | LAR32 | 0.0066 | 0.0000 | 0.0000 |
ORW3 | 0.0066 | 0.0000 | 0.0000 | ||||
VST | 0.0066 | 0.0000 | 0.0000 | ||||
FFLAR4 | 0.0062 | 0.0013 | 0.0000 | ||||
LAR4 | 0.0066 | 0.0000 | 0.0000 |
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Awenlimobor, A.E.; Smith, D.E. Determination of the Steady State Fiber Orientation Tensor States in Homogeneous Flows with Newton–Raphson Iteration Using Exact Jacobians. J. Compos. Sci. 2025, 9, 433. https://doi.org/10.3390/jcs9080433
Awenlimobor AE, Smith DE. Determination of the Steady State Fiber Orientation Tensor States in Homogeneous Flows with Newton–Raphson Iteration Using Exact Jacobians. Journal of Composites Science. 2025; 9(8):433. https://doi.org/10.3390/jcs9080433
Chicago/Turabian StyleAwenlimobor, Aigbe E., and Douglas E. Smith. 2025. "Determination of the Steady State Fiber Orientation Tensor States in Homogeneous Flows with Newton–Raphson Iteration Using Exact Jacobians" Journal of Composites Science 9, no. 8: 433. https://doi.org/10.3390/jcs9080433
APA StyleAwenlimobor, A. E., & Smith, D. E. (2025). Determination of the Steady State Fiber Orientation Tensor States in Homogeneous Flows with Newton–Raphson Iteration Using Exact Jacobians. Journal of Composites Science, 9(8), 433. https://doi.org/10.3390/jcs9080433