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hirish99hirish99
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What is the actual biharmonic 2d green's function?
1 parent 954efe1 commit 6bf06c8

9 files changed

Lines changed: 266 additions & 1653 deletions

‎sumpy/recurrence.py‎

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@@ -277,7 +277,7 @@ def recurrence_from_pde(pde: LinearPDESystemOperator) -> sp.Expr:
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"""
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ode_in_r, var, ode_order = pde_to_ode_in_r(pde)
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ode_in_x = ode_in_r_to_x(ode_in_r, var, ode_order).simplify()
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ode_in_x_cleared = (ode_in_x * var[0]**(ode_order+1)).simplify()
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ode_in_x_cleared = (ode_in_x * var[0]**(pde.order*2-1)).simplify()
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# ode_in_x_cleared shouldn't have rational function coefficients
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assert sp.together(ode_in_x_cleared) == ode_in_x_cleared
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f_x_derivs = _make_sympy_vec("f_x", ode_order+1)

‎test/biharmonic.ipynb‎

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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import sympy as sp\n",
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"\n",
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"from sumpy.expansion.diff_op import (\n",
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" make_identity_diff_op,\n",
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")\n",
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"from collections import namedtuple\n",
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"DerivativeIdentifier = namedtuple(\"DerivativeIdentifier\", [\"mi\", \"vec_idx\"])\n",
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"\n",
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"from sumpy.recurrence import _make_sympy_vec, get_reindexed_and_center_origin_on_axis_recurrence\n",
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"\n",
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"from immutabledict import immutabledict\n",
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"from sumpy.expansion.diff_op import LinearPDESystemOperator"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [],
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"source": [
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"var = _make_sympy_vec(\"x\", 2)\n",
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"var_t = _make_sympy_vec(\"t\", 2)\n",
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"abs_dist = sp.sqrt((var[0]-var_t[0])**2 + (var[1]-var_t[1])**2)\n",
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"w = make_identity_diff_op(2)\n",
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"\n",
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"partial_1x = DerivativeIdentifier((4,0), 0)\n",
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"partial_1y = DerivativeIdentifier((0,4), 0)\n",
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"biharmonic_op = {partial_1x: 1, partial_1y: 1}\n",
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"list_pde = immutabledict(biharmonic_op)\n",
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"\n",
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"biharmonic_pde = LinearPDESystemOperator(2, (list_pde,))\n",
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"g_x_y = abs_dist**2 * (sp.log(abs_dist)-1)\n",
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"\n",
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"n_init, _, r = get_reindexed_and_center_origin_on_axis_recurrence(biharmonic_pde)\n",
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"\n",
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"derivs = [sp.diff(g_x_y,\n",
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" var_t[0], i).subs(var_t[0], 0).subs(var_t[1], 0)\n",
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" for i in range(8)]\n",
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"\n",
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"x_coord = np.random.rand() # noqa: NPY002\n",
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"y_coord = np.random.rand() # noqa: NPY002\n",
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"coord_dict = {var[0]: var[0], var[1]: var[1]}\n",
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"derivs = [d.subs(coord_dict) for d in derivs]\n",
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"\n",
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"n = sp.symbols(\"n\")\n",
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"s = sp.Function(\"s\")\n",
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"\n",
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"# pylint: disable-next=not-callable\n",
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"subs_dict = {s(0): derivs[0], s(1): derivs[1], s(2): derivs[1], s(3): derivs[1]}\n",
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"check = []\n",
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"\n",
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"assert n_init == 4\n",
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"max_order_check = 8\n",
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"for i in range(n_init, max_order_check):\n",
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" check.append(r.subs(n, i).subs(subs_dict) - derivs[i])\n",
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" # pylint: disable-next=not-callable\n",
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" subs_dict[s(i)] = derivs[i]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [],
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"source": [
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"max_order_check = 8\n",
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"for i in range(n_init, max_order_check):\n",
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" check.append(r.subs(n, i).subs(subs_dict) - derivs[i])\n",
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" # pylint: disable-next=not-callable\n",
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" subs_dict[s(i)] = derivs[i]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/latex": [
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"$\\displaystyle 16.9964566798618$"
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],
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"text/plain": [
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"16.9964566798618"
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]
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},
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"execution_count": 14,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"r.subs(n, 4).subs(subs_dict).subs({var[0]: 1.2, var[1]: 2.3})"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"{s(0): (x0**2 + x1**2)*(log(sqrt(x0**2 + x1**2)) - 1),\n",
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" s(1): -2*x0*(log(sqrt(x0**2 + x1**2)) - 1) - x0,\n",
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" s(2): -2*x0*(log(sqrt(x0**2 + x1**2)) - 1) - x0,\n",
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" s(3): -2*x0*(log(sqrt(x0**2 + x1**2)) - 1) - x0,\n",
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" s(4): 2*(-24*x0**4/(x0**2 + x1**2)**2 + 8*x0**2*(4*x0**2/(x0**2 + x1**2) - 3)/(x0**2 + x1**2) + 12*x0**2/(x0**2 + x1**2) + 3)/(x0**2 + x1**2),\n",
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" s(5): -4*x0*(-24*x0**4/(x0**2 + x1**2)**2 + 40*x0**2/(x0**2 + x1**2) - 15)/(x0**2 + x1**2)**2,\n",
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" s(6): 12*(-320*x0**6/(x0**2 + x1**2)**3 + 360*x0**4/(x0**2 + x1**2)**2 + 24*x0**2*(16*x0**4/(x0**2 + x1**2)**2 - 20*x0**2/(x0**2 + x1**2) + 5)/(x0**2 + x1**2) - 60*x0**2/(x0**2 + x1**2) - 5)/(x0**2 + x1**2)**2,\n",
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" s(7): -48*x0*(-160*x0**6/(x0**2 + x1**2)**3 + 336*x0**4/(x0**2 + x1**2)**2 - 210*x0**2/(x0**2 + x1**2) + 35)/(x0**2 + x1**2)**3}"
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]
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},
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"subs_dict"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "inteq",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.9"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}

‎test/gridfree_taylor_recurrence.ipynb‎

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