add exponential case with 3-point Gauss rule and update README

examples/Freefem/MMS/1D/README.md

@@ -54,6 +54,10 @@ The rule is determined by the choice of $n_g$, $\xi_k$, and $w_k$:
 |------|--------|----------|--------|
 | 1-point (midpoint) | 1 | $0$ | $2$ |
 | 2-point | 2 | $\pm \frac{1}{\sqrt{3}}$ | $1$ |
+| 3-point | 3 | $0,\ \pm\sqrt{\frac{3}{5}}$ | $\frac{8}{9}\ (k=2),\quad \frac{5}{9}\ (k=1,3)$ |
+
+> **3-point rule:** $\xi_1 = -\sqrt{\tfrac{3}{5}},\ \xi_2 = 0,\ \xi_3 = \sqrt{\tfrac{3}{5}}$;
+> $w_1 = w_3 = \tfrac{5}{9},\ w_2 = \tfrac{8}{9}$
 
 ### Body Force
 
@@ -206,3 +210,48 @@ With a 2-point quadrature rule:
 $$
 F_0 = \frac{h}{6}\left[2f(x_0) + f(x_1)\right], \quad F_i = h\, f(x_i), \quad F_N = \frac{h}{6}\left[f(x_{N-1}) + 2f(x_N)\right]
 $$
+
+$$
+F_i^{(e)} = \frac{h}{2} \sum_{k=1}^{2} w_k\, f(x_k^{(e)})\, \phi_i(x_k^{(e)})
+$$
+
+---
+## Case 4: Exponential Function
+
+### Manufactured Solution
+
+
+$$u_{ex}(x) = e^x - 1$$
+
+### Source Term
+
+$$
+f(x)=−E exf(x) = -E\, e^xf(x)=−Eex
+$$
+
+### Boundary Conditions
+
+
+$$ 
+u(0)=0
+$$
+
+
+$$
+\left.\frac{du}{dx}\right|_{1} = e
+$$
+
+Neumann force:
+
+$$
+F_N = E\, e
+$$
+
+
+### Source Term Discretization
+
+With a 3-point quadrature rule:
+
+$$
+F_i^{(e)} = \frac{h}{2} \sum_{k=1}^{3} w_k\, f(x_k^{(e)})\, \phi_i(x_k^{(e)})
+$$

examples/Freefem/MMS/1D/exponential.py

@@ -0,0 +1,38 @@
+"""
+Exponential MMS (non-dimensional, x = x_dim/L ∈ [0,1], E = E_dim/L):
+
+    u_ex(x) = exp(x) - 1              du/dx   = exp(x)
+
+    f(x)    = -E * exp(x)     
+BC:
+    u(0)   = 0     (Dirichlet)
+    u'(1)  = e     (Neumann)  =>  F_N = E·u'(1) = E·e
+"""
+from manufactured_solution import MMSCase1D
+from bar import load_params, line_quadrature, build_bar_scene, run_scene
+import numpy as np
+
+class Exponential(MMSCase1D):
+    name       = "exponential"
+    plot_label = r"$e^x - 1$"
+    quadrature = staticmethod(line_quadrature(3)) # using 3 pts quadrature 
+
+    def u_ex(self, xi):
+        return np.exp(xi) - 1.0
+
+    def du_ex(self, xi):
+        return np.exp(xi)
+
+    def source(self, xi, E):
+        return -E * np.exp(xi)   # f = -E·u'' = -E·exp(x)
+
+mms = Exponential()
+
+def createScene(rootNode):
+    cfg  = load_params()
+    L, E = cfg["length"], cfg["youngModulus"]
+    build_bar_scene(rootNode, mms, E / L, cfg["nx"])
+    return rootNode
+
+if __name__ == "__main__":
+    run_scene(mms)

examples/Freefem/MMS/1D/params.json

@@ -4,7 +4,8 @@
     "nx": 10,
     "nxConvergence": {
         "quadratic":  [2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 32, 64],
-        "cubic":      [2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 32, 64],
-        "sinusoidal": [10, 20, 40, 80, 160]
+        "cubic":      [3, 4, 5, 6, 8, 10, 12, 16, 20, 32, 64],
+        "sinusoidal": [10, 20, 40, 80, 160],
+        "exponential" :[10, 20, 40, 80, 160]
     }
 }

examples/Freefem/MMS/1D/run_convergence.py

@@ -7,6 +7,7 @@
 from quadratic  import mms as quadratic_mms
 from cubic      import mms as cubic_mms
 from sinusoidal import mms as sinusoidal_mms
+from exponential import mms as exponential_mms 
 
 from bar import (
     RESULTS_DIR,
@@ -112,5 +113,5 @@ def run(mms, L, E, nx_list):
 if __name__ == "__main__":
     cfg  = load_params()
     L, E = cfg["length"], cfg["youngModulus"]
-    for mms in (quadratic_mms, cubic_mms, sinusoidal_mms):
+    for mms in (quadratic_mms, cubic_mms, sinusoidal_mms, exponential_mms):
         run(mms, L, E, cfg["nxConvergence"][mms.name])

examples/Freefem/MMS/fem.py

@@ -15,6 +15,8 @@
         np.array([2.0])),
     2: (np.array([-1.0 / np.sqrt(3.0), 1.0 / np.sqrt(3.0)]),
         np.array([1.0, 1.0])),
+    3: (np.array([-np.sqrt(3.0 / 5.0), 0.0, np.sqrt(3.0 / 5.0)]),
+        np.array([5.0 / 9.0, 8.0 / 9.0, 5.0 / 9.0])),
 }