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base_functions.h
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1/** \file base_functions.c
2
3*/
4
5#ifndef __BASE_FUNCTIONS_H__
6#define __BASE_FUNCTIONS_H__
7
8#ifdef __cplusplus
9extern "C" {
10#endif
11
12/**
13\brief Calculate Legendre approximation basis
14
15\ingroup mofem_base_functions
16
17Lagrange polynomial is given by
18\f[
20\f]
21and following terms are generated inductively
22\f[
23L_{l+1}=\frac{2l+1}{l+1}sL_l(s)-\frac{l}{l+1}L_{l-1}(s)
24\f]
25
26Note that:
27\f[
29\f]
30where \f$\xi_i\f$ are barycentric coordinates of element.
31
32\param p is approximation order
33\param s is position \f$s\in[-1,1]\f$
34\param diff_s derivatives of shape functions, i.e. \f$\frac{\partial s}{\partial 35\xi_i}\f$ \retval L approximation functions \retval diffL derivatives, i.e.
36\f$\frac{\partial L}{\partial \xi_i}\f$ \param dim dimension \return error code
37
38*/
39PetscErrorCode Legendre_polynomials(int p, double s, double *diff_s, double *L,
40 double *diffL, const int dim);
41
42/**
43\brief Calculate Jacobi approximation basis
44
45For more details see \cite fuentes2015orientation
46
47\param p is approximation order
48\param alpha polynomial parameter
49\param x is position \f$s\in[0,t]\f$
50\param t range of polynomial
51\param diff_x derivatives of shape functions, i.e. \f$\frac{\partial x}{\partial 52\xi_i}\f$ \param diff_t derivatives of shape functions, i.e. \f$\frac{\partial 53t}{\partial \xi_i}\f$ \retval L approximation functions \retval diffL
54derivatives, i.e. \f$\frac{\partial L}{\partial \xi_i}\f$ \param dim dimension
55\return error code
56
57*/
58PetscErrorCode Jacobi_polynomials(int p, double alpha, double x, double t,
59 double *diff_x, double *diff_t, double *L,
60 double *diffL, const int dim);
61
62/**
63\brief Calculate integrated Jacobi approximation basis
64
65For more details see \cite fuentes2015orientation
66
67\param p is approximation order
68\param alpha polynomial parameter
69\param x is position \f$s\in[0,t]\f$
70\param t range of polynomial
71\param diff_x derivatives of shape functions, i.e. \f$\frac{\partial x}{\partial 72\xi_i}\f$ \param diff_t derivatives of shape functions, i.e. \f$\frac{\partial 73t}{\partial \xi_i}\f$ \retval L approximation functions \retval diffL
74derivatives, i.e. \f$\frac{\partial L}{\partial \xi_i}\f$ \param dim dimension
75\return error code
76
77*/
78PetscErrorCode IntegratedJacobi_polynomials(int p, double alpha, double x,
79 double t, double *diff_x,
80 double *diff_t, double *L,
81 double *diffL, const int dim);
82
83/**
84 \brief Calculate Lobatto base functions \cite FUENTES2015353.
85
86 \ingroup mofem_base_functions
87
88 Order of first function is 2 and goes to p.
89
90 \param p is approximation order
91 \param s is a mapping of coordinates of edge to \f$[-1, 1]\f$, i.e., \f$s(\xi_1,\cdot,\xi_{dim})\in[-1,1]\f$
92 \param diff_s jacobian of the transformation, i.e. \f$\frac{\partial 93 s}{\partial \xi_i}\f$
94 - output
95 \retval L values basis functions at s
96 \retval diffL derivatives of basis functions at s, i.e. \f$\frac{\partial L}{\partial \xi_i}\f$ \param dim dimension
97 \return error code
98*/
99PetscErrorCode Lobatto_polynomials(int p, double s, double *diff_s, double *L,
100 double *diffL, const int dim);
101
102/**
103 \brief Calculate Kernel Lobatto base functions.
104
105 \ingroup mofem_base_functions
106
107 This is implemented using definitions from Hermes2d
108 <https://github.com/hpfem/hermes> following book by Pavel Solin et al \cite
109 solin2003higher.
110
111 \param p is approximation order
112 \param s is position \f$s\in[-1,1]\f$
113 \param diff_s derivatives of shape functions, i.e. \f$\frac{\partial 114 s}{\partial \xi_i}\f$ \retval L approximation functions \retval diffL
115 derivatives, i.e. \f$\frac{\partial L}{\partial \xi_i}\f$ \param dim dimension
116 \return error code
117
118*/
119PetscErrorCode LobattoKernel_polynomials(int p, double s, double *diff_s,
120 double *L, double *diffL,
121 const int dim);
122
123/// Definitions taken from Hermes2d code
124
125/// kernel functions for Lobatto base
126#define LOBATTO_PHI0(x) (-2.0 * 1.22474487139158904909864203735)
127#define LOBATTO_PHI1(x) (-2.0 * 1.58113883008418966599944677222 * (x))
128#define LOBATTO_PHI2(x) \
129 (-1.0 / 2.0 * 1.87082869338697069279187436616 * (5 * (x) * (x)-1))
130#define LOBATTO_PHI3(x) \
131 (-1.0 / 2.0 * 2.12132034355964257320253308631 * (7 * (x) * (x)-3) * (x))
132#define LOBATTO_PHI4(x) \
133 (-1.0 / 4.0 * 2.34520787991171477728281505677 * \
134 (21 * (x) * (x) * (x) * (x)-14 * (x) * (x) + 1))
135#define LOBATTO_PHI5(x) \
136 (-1.0 / 4.0 * 2.54950975679639241501411205451 * \
137 ((33 * (x) * (x)-30) * (x) * (x) + 5) * (x))
138#define LOBATTO_PHI6(x) \
139 (-1.0 / 32.0 * 2.73861278752583056728484891400 * \
140 (((429 * (x) * (x)-495) * (x) * (x) + 135) * (x) * (x)-5))
141#define LOBATTO_PHI7(x) \
142 (-1.0 / 32.0 * 2.91547594742265023543707643877 * \
143 (((715 * (x) * (x)-1001) * (x) * (x) + 385) * (x) * (x)-35) * (x))
144#define LOBATTO_PHI8(x) \
145 (-1.0 / 64.0 * 3.08220700148448822512509619073 * \
146 ((((2431 * (x) * (x)-4004) * (x) * (x) + 2002) * (x) * (x)-308) * (x) * \
147 (x) + \
148 7))
149#define LOBATTO_PHI9(x) \
150 (-1.0 / 128.0 * 6.4807406984078603784382721642 * \
151 ((((4199 * (x) * (x)-7956) * (x) * (x) + 4914) * (x) * (x)-1092) * (x) * \
152 (x) + \
153 63) * \
154 (x))
155
156/// Derivatives of kernel functions for Lobbatto base
157#define LOBATTO_PHI0X(x) (0)
158#define LOBATTO_PHI1X(x) (-2.0 * 1.58113883008418966599944677222)
159#define LOBATTO_PHI2X(x) \
160 (-1.0 / 2.0 * 1.87082869338697069279187436616 * (10 * (x)))
161#define LOBATTO_PHI3X(x) \
162 (-1.0 / 2.0 * 2.12132034355964257320253308631 * (21.0 * (x) * (x)-3.0))
163#define LOBATTO_PHI4X(x) \
164 (-1.0 / 4.0 * 2.34520787991171477728281505677 * \
165 ((84.0 * (x) * (x)-28.0) * (x)))
166#define LOBATTO_PHI5X(x) \
167 (-1.0 / 4.0 * 2.54950975679639241501411205451 * \
168 ((165.0 * (x) * (x)-90.0) * (x) * (x) + 5.0))
169#define LOBATTO_PHI6X(x) \
170 (-1.0 / 32.0 * 2.73861278752583056728484891400 * \
171 (((2574.0 * (x) * (x)-1980.0) * (x) * (x) + 270.0) * (x)))
172#define LOBATTO_PHI7X(x) \
173 (-1.0 / 32.0 * 2.91547594742265023543707643877 * \
174 (((5005.0 * (x) * (x)-5005.0) * (x) * (x) + 1155.0) * (x) * (x)-35.0))
175#define LOBATTO_PHI8X(x) \
176 (-1.0 / 64.0 * 3.08220700148448822512509619073 * \
177 ((((19448.0 * (x) * (x)-24024.0) * (x) * (x) + 8008.0) * (x) * (x)-616.0) * \
178 (x)))
179#define LOBATTO_PHI9X(x) \
180 (-1.0 / 128.0 * 6.4807406984078603784382721642 * \
181 ((((37791.0 * (x) * (x)-55692.0) * (x) * (x) + 24570.0) * (x) * \
182 (x)-3276.0) * \
183 (x) * (x)-63.0))
184
185#ifdef __cplusplus
186}
187#endif
188
189#endif //__BASE_FUNCTIONS_H__
static Index< 'L', 3 > L
static Index< 'p', 3 > p
PetscErrorCode IntegratedJacobi_polynomials(int p, double alpha, double x, double t, double *diff_x, double *diff_t, double *L, double *diffL, const int dim)
Calculate integrated Jacobi approximation basis.
PetscErrorCode Jacobi_polynomials(int p, double alpha, double x, double t, double *diff_x, double *diff_t, double *L, double *diffL, const int dim)
Calculate Jacobi approximation basis.
const int dim
PetscErrorCode Legendre_polynomials(int p, double s, double *diff_s, double *L, double *diffL, const int dim)
Calculate Legendre approximation basis.
PetscErrorCode Lobatto_polynomials(int p, double s, double *diff_s, double *L, double *diffL, const int dim)
Calculate Lobatto base functions .
PetscErrorCode LobattoKernel_polynomials(int p, double s, double *diff_s, double *L, double *diffL, const int dim)
Calculate Kernel Lobatto base functions.
constexpr double t
plate stiffness
Definition: plate.cpp:59