Merge pull request #38 from adtzlr/fix-microsphere-langevin

fix langevin chain model

README.md

@@ -110,9 +110,9 @@ Available [anisotropic hyperelastic material models](https://github.com/adtzlr/m
 
 Available [micro-sphere hyperelastic material models](https://github.com/adtzlr/matadi/blob/main/matadi/models/microsphere) (Miehe, Göktepe, Lulei) [[2](https://doi.org/10.1016/j.jmps.2004.03.011)]:
 - [affine stretch](https://doi.org/10.1016/j.jmps.2004.03.011) ([code](https://github.com/adtzlr/matadi/blob/main/matadi/models/microsphere/affine/_models.py#L5-L15))
-- [affine tube](https://doi.org/10.1016/j.jmps.2004.03.011) ([code](https://github.com/adtzlr/matadi/blob/main/matadi/models/microsphere/affine/_models.py#L18-L28))
+- [affine tube](https://doi.org/10.1016/j.jmps.2004.03.011) ([code](https://github.com/adtzlr/matadi/blob/main/matadi/models/microsphere/affine/_models.py#L18-L29))
 - [non-affine stretch](https://doi.org/10.1016/j.jmps.2004.03.011) ([code](https://github.com/adtzlr/matadi/blob/main/matadi/models/microsphere/nonaffine/_models.py#L5-L15))
-- [non-affine tube](https://doi.org/10.1016/j.jmps.2004.03.011) ([code](https://github.com/adtzlr/matadi/blob/main/matadi/models/microsphere/nonaffine/_models.py#L18-L28))
+- [non-affine tube](https://doi.org/10.1016/j.jmps.2004.03.011) ([code](https://github.com/adtzlr/matadi/blob/main/matadi/models/microsphere/nonaffine/_models.py#L18-L29))
 
 Any user-defined isotropic hyperelastic strain energy density function may be passed as the `fun` argument of `MaterialHyperelastic` by using the following template:
 

matadi/__about__.py

@@ -1 +1 @@
-__version__ = "0.0.15"
+__version__ = "0.0.16"

matadi/models/microsphere/__init__.py

@@ -3,5 +3,6 @@
 from . import quadrature
 from ._chain import (
     langevin,
+    langevin2,
     gauss,
 )

matadi/models/microsphere/_chain.py

@@ -1,11 +1,28 @@
-from ...math import sqrt, atanh
+from ...math import log, sinh, sqrt
 
 
 def langevin(stretch, mu, N):
-    """Langevin model (Padé approximation) given by the free energy
-    of a single chain as a function of the stretch."""
+    """Langevin model given by the free energy of a single chain as a function
+    of the stretch (assuming a complex valued logarithm). The inverse
+    Langevin function is defined by a Padé approximation."""
+
+    x = stretch / sqrt(N)
+    L = x * (3 - x ** 2) / (1 - x ** 2)
+
+    return mu * N * (x * L + log(L / sinh(L)))
+
+
+def langevin2(stretch, mu, N):
+    """Langevin model given by the free energy of a single chain as a function
+    of the stretch (assuming a complex valued logarithm). The inverse Langevin
+    function is defined by a Padé approximation.
+
+    **Note**: This function is optimized for fast gradient evaluation but
+    cannot be used to calculate the actual free energy - only its derivative
+    w.r.t. the stretch gives a real-valued result for the region of interest
+    `N > stretch ** 2`."""
 
-    return mu * (stretch + 2 * sqrt(N) * atanh(stretch / sqrt(N)))
+    return mu * (stretch ** 2 / 2 - N * log(stretch ** 2 - N))
 
 
 def gauss(stretch, mu):

pyproject.toml

@@ -1,6 +1,6 @@
 [project]
 name = "matadi"
-version = "0.0.15"
+version = "0.0.16"
 description = "Material Definition with Automatic Differentiation"
 readme = "README.md"
 requires-python = ">=3.6"

setup.cfg

@@ -1,6 +1,6 @@
 [metadata]
 name = matadi
-version = 0.0.15
+version = 0.0.16
 author = Andreas Dutzler
 author_email = [email protected]
 description = Material Definition with Automatic Differentiation

tests/test_models.py

@@ -29,32 +29,21 @@ def library():
         md.arruda_boyce: {"C1": 1.0, "limit": 3.2},
         md.extended_tube: {"Gc": 0.1867, "Ge": 0.2169, "beta": 0.2, "delta": 0.09693},
         md.van_der_waals: {"mu": 1.0, "beta": 0.1, "a": 0.5, "limit": 5.0},
-        md.microsphere.affine.stretch: {
-            "quadrature": q,
-            "f": md.microsphere.gauss,
-            "kwargs": {"mu": 1},
-        },
-        md.microsphere.nonaffine.stretch: {
-            "quadrature": q,
-            "p": 2.7,
-            "f": md.microsphere.gauss,
-            "kwargs": {"mu": 1},
-        },
         md.microsphere.affine.stretch: {
             "quadrature": q,
             "f": md.microsphere.langevin,
             "kwargs": {"mu": 1, "N": 10},
         },
         md.microsphere.affine.tube: {
             "quadrature": q,
-            "f": md.microsphere.langevin,
+            "f": md.microsphere.langevin2,
             "kwargs": {"mu": 1, "N": 10},
         },
         md.microsphere.nonaffine.stretch: {
             "quadrature": q,
             "p": 2.7,
-            "f": md.microsphere.langevin,
-            "kwargs": {"mu": 1, "N": 10},
+            "f": md.microsphere.gauss,
+            "kwargs": {"mu": 1},
         },
         md.microsphere.nonaffine.tube: {
             "quadrature": q,