diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index 4232f7a..ae2d597 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-10-13T13:50:42","documenter_version":"1.1.1"}} \ No newline at end of file +{"documenter":{"julia_version":"1.10.0","generation_timestamp":"2024-01-15T15:41:45","documenter_version":"1.1.1"}} \ No newline at end of file diff --git a/dev/index.html b/dev/index.html index 55f5ceb..29be364 100644 --- a/dev/index.html +++ b/dev/index.html @@ -15,4 +15,4 @@ lines!(ax, y, rel_error, label = string(f)) end axislegend(position = :lt)

example

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+ diff --git a/dev/methods/index.html b/dev/methods/index.html index 2f33c08..d0b2820 100644 --- a/dev/methods/index.html +++ b/dev/methods/index.html @@ -3,11 +3,11 @@ y, m::InverseLangevinApproximations.AbstractInverseLangevinApproximation ) -> Any -

Method for computing the inverse Langevin function.

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Approximations

InverseLangevinApproximations.BergstromApproximation — Type

From Ref. [1]:

\[\mathcal{L}^{-1}(y) \approx\begin{cases} +

Method for computing the inverse Langevin function.

  • y: The value of the Langevin Function
  • m: The approximation Method
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Approximations

InverseLangevinApproximations.BergstromApproximation — Type

From Ref. [1]:

\[\mathcal{L}^{-1}(y) \approx\begin{cases} 1.31446\tan{1.58986y}+0.91209y & |y|\leq 0.84136 \\ \frac{1}{\text{sign}(y)-y} & 0.84136 \leq |x| < 1.0 \\ - \end{cases}\]

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InverseLangevinApproximations.CohenExact3_2 — Type

From Ref. [2]:

\[\mathcal{L}^{-1}(y) \approx y\frac{3-\frac{36}{35}y^2}{1-\frac{33}{35}y^2}\]

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InverseLangevinApproximations.CohenRounded3_2 — Type

From Ref. [2]:

\[\mathcal{L}^{-1}(y) \approx y\frac{3-y^2}{1-y^2}\]

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InverseLangevinApproximations.DarabiItskov — Type

From Ref. [3]:

\[{\mathcal{L}^{- 1}}(y) \approx y\frac{{y^{2} - 3y + 3}}{{1 - y}}.\]

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InverseLangevinApproximations.Jedynak2017 — Type

From Ref. [4]:

\[\mathcal{L}^{-1}(y) \approx y\frac{3-\frac{773}{768}y^2-\frac{1300}{1351}y^4+\frac{501}{340}y^6-\frac{678}{138}y^8}{(1-y)(1+\frac{866}{853}*y)}\]

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InverseLangevinApproximations.KuhnGrunApproximation — Type

From Ref. [5]:

\[\mathcal{L}^{-1}(y) \approx 3y + \frac{9}{5}y^3 + \frac{297}{175}y^5 + \frac{1539}{875}y^7 + \frac{126117}{67375}y^9 + \frac{43733439}{21896875}y^{11} + \frac{231321177}{109484375}y^{13} + \frac{20495009043}{9306171875}y^{15} + \frac{1073585186448381}{476522530859375}y^{17} + \frac{4387445039583}{1944989921875}y^{19}\]

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InverseLangevinApproximations.NguessongBedaPeyraut — Type

From Ref. [6]:

\[{\mathcal{L}^{-1}}\left( y \right)\approx y\frac{3-y^{2}}{1-y^{2}}-0.488y^{3.243}+3.311y^{4.789}\left( y-0.76 \right)\left( y-1 \right)\]

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InverseLangevinApproximations.PadeApproximation3_2 — Type

From Ref. [2]:

\[\mathcal{L}^{-1}(y) \approx y \frac{(3 - \frac{36}{35} y^2)}{(1 - \frac{33}{35} y^2)}\]

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InverseLangevinApproximations.PadeApproximation_1_2 — Type

From Ref. [7]:

\[\mathcal{L}^{-1}(y) \approx \frac{3y}{1-\frac{3}{5}y^2}\]

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InverseLangevinApproximations.PadeApproximation_1_4 — Type

From Ref. [7]:

\[\mathcal{L}^{-1}(y) \approx \frac{3y}{1-\frac{3}{5}y^2-\frac{36}{175}y^4}\]

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InverseLangevinApproximations.PadeApproximation_3_0 — Type

From Ref. [7]:

\[\mathcal{L}^{-1}(y) \approx 3y + \frac{9}{5}y^3\]

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InverseLangevinApproximations.PadeApproximation_5_0 — Type

From Ref. [7]:

\[\mathcal{L}^{-1}(y) \approx 3y + \frac{9}{5}y^3 + \frac{297}{175}y^5\]

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InverseLangevinApproximations.PusoApproximation — Type

From Ref. [8]:

\[\mathcal{L}^{-1}(y) \approx \frac{3y}{1-y^3}\]

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InverseLangevinApproximations.TreloarApproximation — Type

From Ref. [9]:

\[\mathcal{L}^{-1}(y) \approx \frac{3 y}{(1 - (\frac{3}{5} y^2 + \frac{36}{175} y^4 + \frac{108}{875} y^6))}\]

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InverseLangevinApproximations.WarnerApproximation — Type

From Ref. [10]:

\[\mathcal{L}^{-1}(y) \approx \frac{3y}{1-y^2}\]

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References

[1]
+ \end{cases}\]

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InverseLangevinApproximations.CohenExact3_2 — Type

From Ref. [2]:

\[\mathcal{L}^{-1}(y) \approx y\frac{3-\frac{36}{35}y^2}{1-\frac{33}{35}y^2}\]

source
InverseLangevinApproximations.CohenRounded3_2 — Type

From Ref. [2]:

\[\mathcal{L}^{-1}(y) \approx y\frac{3-y^2}{1-y^2}\]

source
InverseLangevinApproximations.DarabiItskov — Type

From Ref. [3]:

\[{\mathcal{L}^{- 1}}(y) \approx y\frac{{y^{2} - 3y + 3}}{{1 - y}}.\]

source
InverseLangevinApproximations.Jedynak2017 — Type

From Ref. [4]:

\[\mathcal{L}^{-1}(y) \approx y\frac{3-\frac{773}{768}y^2-\frac{1300}{1351}y^4+\frac{501}{340}y^6-\frac{678}{138}y^8}{(1-y)(1+\frac{866}{853}*y)}\]

source
InverseLangevinApproximations.KuhnGrunApproximation — Type

From Ref. [5]:

\[\mathcal{L}^{-1}(y) \approx 3y + \frac{9}{5}y^3 + \frac{297}{175}y^5 + \frac{1539}{875}y^7 + \frac{126117}{67375}y^9 + \frac{43733439}{21896875}y^{11} + \frac{231321177}{109484375}y^{13} + \frac{20495009043}{9306171875}y^{15} + \frac{1073585186448381}{476522530859375}y^{17} + \frac{4387445039583}{1944989921875}y^{19}\]

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InverseLangevinApproximations.NguessongBedaPeyraut — Type

From Ref. [6]:

\[{\mathcal{L}^{-1}}\left( y \right)\approx y\frac{3-y^{2}}{1-y^{2}}-0.488y^{3.243}+3.311y^{4.789}\left( y-0.76 \right)\left( y-1 \right)\]

source
InverseLangevinApproximations.PadeApproximation3_2 — Type

From Ref. [2]:

\[\mathcal{L}^{-1}(y) \approx y \frac{(3 - \frac{36}{35} y^2)}{(1 - \frac{33}{35} y^2)}\]

source
InverseLangevinApproximations.PadeApproximation_1_2 — Type

From Ref. [7]:

\[\mathcal{L}^{-1}(y) \approx \frac{3y}{1-\frac{3}{5}y^2}\]

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InverseLangevinApproximations.PadeApproximation_1_4 — Type

From Ref. [7]:

\[\mathcal{L}^{-1}(y) \approx \frac{3y}{1-\frac{3}{5}y^2-\frac{36}{175}y^4}\]

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InverseLangevinApproximations.PadeApproximation_3_0 — Type

From Ref. [7]:

\[\mathcal{L}^{-1}(y) \approx 3y + \frac{9}{5}y^3\]

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InverseLangevinApproximations.PadeApproximation_5_0 — Type

From Ref. [7]:

\[\mathcal{L}^{-1}(y) \approx 3y + \frac{9}{5}y^3 + \frac{297}{175}y^5\]

source
InverseLangevinApproximations.PusoApproximation — Type

From Ref. [8]:

\[\mathcal{L}^{-1}(y) \approx \frac{3y}{1-y^3}\]

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InverseLangevinApproximations.TreloarApproximation — Type

From Ref. [9]:

\[\mathcal{L}^{-1}(y) \approx \frac{3 y}{(1 - (\frac{3}{5} y^2 + \frac{36}{175} y^4 + \frac{108}{875} y^6))}\]

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InverseLangevinApproximations.WarnerApproximation — Type

From Ref. [10]:

\[\mathcal{L}^{-1}(y) \approx \frac{3y}{1-y^2}\]

source

References

[1]
J. S. Bergstr{\"o}m. Large strain time-dependent behavior of elastomeric materials. Phd thesis, Massachusetts Institute of Technology (1999).
[2]
@@ -38,4 +38,4 @@
H. R. Warner Jr. Kinetic theory and rheology of dilute suspensions of finitely extendible dumbbells. Industrial \& Engineering Chemistry Fundamentals 11, 379–387 (1972).
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