BELFEM 0.9.0
Berkeley Lab Finite Element Framework
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cl_IF_QUAD9.hpp
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1/*
2 * BELFEM -- The Berkeley Lab Finite Element Framework
3 * Copyright (c) 2026, The Regents of the University of California,
4 * through Lawrence Berkeley National Laboratory (subject to receipt of any required
5 * approvals from the U.S. Dept. of Energy). All rights reserved.
6 *
7 * Developers: Christian Messe, Gregory Giard
8 *
9 * See the top-level LICENSE file for the complete license and disclaimer.
10 */
11
12#ifndef BELFEM_CL_IF_QUAD9_HPP
13#define BELFEM_CL_IF_QUAD9_HPP
14
15#include "cl_IF_InterpolationFunctionTemplate.hpp"
16
17namespace belfem
18{
19 namespace fem
20 {
21//------------------------------------------------------------------------------
22
23 template<>
24 InterpolationOrder
25 InterpolationFunctionTemplate<
26 GeometryType::QUAD, InterpolationType::LAGRANGE, 2, 9 >
31
32//------------------------------------------------------------------------------
33
34 template<>
35 ElementType
36 InterpolationFunctionTemplate<
37 GeometryType::QUAD, InterpolationType::LAGRANGE, 2, 9 >
38 ::element_type() const
39 {
40 return ElementType::QUAD9;
41 }
42
43//------------------------------------------------------------------------------
44
45 template<>
46 void
47 InterpolationFunctionTemplate<
48 GeometryType::QUAD, InterpolationType::LAGRANGE, 2, 9 >
49 ::param_coords( Matrix< real > & aXiHat ) const
50 {
51 aXiHat.set_size( 2, 9 );
52
53 aXiHat( 0, 0 ) = -1.000000;
54 aXiHat( 1, 0 ) = -1.000000;
55
56 aXiHat( 0, 1 ) = 1.000000;
57 aXiHat( 1, 1 ) = -1.000000;
58
59 aXiHat( 0, 2 ) = 1.000000;
60 aXiHat( 1, 2 ) = 1.000000;
61
62 aXiHat( 0, 3 ) = -1.000000;
63 aXiHat( 1, 3 ) = 1.000000;
64
65 aXiHat( 0, 4 ) = 0.000000;
66 aXiHat( 1, 4 ) = -1.000000;
67
68 aXiHat( 0, 5 ) = 1.000000;
69 aXiHat( 1, 5 ) = 0.000000;
70
71 aXiHat( 0, 6 ) = 0.000000;
72 aXiHat( 1, 6 ) = 1.000000;
73
74 aXiHat( 0, 7 ) = -1.000000;
75 aXiHat( 1, 7 ) = 0.000000;
76
77 aXiHat( 0, 8 ) = 0.000000;
78 aXiHat( 1, 8 ) = 0.000000;
79 }
80
81//------------------------------------------------------------------------------
82
83 template<>
84 void
85 InterpolationFunctionTemplate<
86 GeometryType::QUAD, InterpolationType::LAGRANGE, 2, 9 >::N(
87 const Vector< real > & aXi,
88 Matrix< real > & aN ) const
89 {
90 const real xi = aXi( 0 );
91 const real eta = aXi( 1 );
92
93 const real c = xi * eta * 0.25;
94 const real xi2 = xi*xi;
95 const real eta2 = eta*eta;
96
97 aN.set_size( 1, 9 );
98
99 aN( 0, 0 ) = ( c * ( eta - 1.0 ) * (xi - 1.0) );
100 aN( 0, 1 ) = ( c * ( eta - 1.0 ) * (xi + 1.0) );
101 aN( 0, 2 ) = ( c * ( eta + 1.0 ) * (xi + 1.0) );
102 aN( 0, 3 ) = ( c * ( eta + 1.0 ) * (xi - 1.0) );
103 aN( 0, 4 ) = ( eta * ( 1.0 - xi2 ) * ( eta - 1.0 ) ) * 0.5;
104 aN( 0, 5 ) = ( xi * ( 1.0 - eta2)*( xi + 1.0 ) )*0.5;
105 aN( 0, 6 ) = ( eta * (1.0 - xi2)*( eta + 1.0 ) )*0.5;
106 aN( 0, 7 ) = ( xi*( 1.0 - eta2 )*( xi - 1.0 ) )*0.5;
107 aN( 0, 8 ) = ( eta2 - 1.0 )*( xi2 - 1.0 );
108 }
109
110//------------------------------------------------------------------------------
111
112 template<>
113 void
114 InterpolationFunctionTemplate<
115 GeometryType::QUAD, InterpolationType::LAGRANGE, 2, 9 >::dNdXi(
116 const Vector< real > & aXi,
117 Matrix< real > & adNdXi ) const
118 {
119 const real xi = aXi( 0 );
120 const real eta = aXi( 1 );
121
122 // often used constants
123 const real c = xi*eta;
124 const real xi2 = xi*xi;
125 const real eta2 = eta*eta;
126
127 adNdXi.set_size( 2, 9 );
128
129 adNdXi( 0, 0 ) = ( eta * ( 2.0 * xi - 1.0 ) * ( eta - 1.0 ) ) * 0.25;
130 adNdXi( 1, 0 ) = ( xi * ( 2.0 * eta - 1.0 ) * ( xi - 1.0 ) ) * 0.25;
131
132 adNdXi( 0, 1 ) = ( eta * ( 2.0 * xi + 1.0 ) * ( eta - 1.0 ) ) * 0.25;
133 adNdXi( 1, 1 ) = ( xi * ( 2.0 * eta - 1.0 ) * ( xi + 1.0 ) ) * 0.25;
134
135 adNdXi( 0, 2 ) = ( eta * ( 2.0 * xi + 1.0 ) * ( eta + 1.0 ) ) * 0.25;
136 adNdXi( 1, 2 ) = ( xi * ( 2.0 * eta + 1.0 ) * ( xi + 1.0 ) ) * 0.25;
137
138 adNdXi( 0, 3 ) = ( eta * ( 2.0 * xi - 1.0 ) * ( eta + 1.0 ) ) * 0.25;
139 adNdXi( 1, 3 ) = ( xi * ( 2.0 * eta + 1.0 ) * ( xi - 1.0 ) ) * 0.25;
140
141 adNdXi( 0, 4 ) = - c * ( eta - 1.0 );
142 adNdXi( 1, 4 ) = -( ( 2.0 * eta - 1.0 ) * ( xi2 - 1.0 ) ) * 0.5;
143
144 adNdXi( 0, 5 ) = -( ( eta2 - 1.0 ) * ( 2.0 * xi + 1.0 ) ) * 0.5;
145 adNdXi( 1, 5 ) = - c * ( xi + 1.0 );
146
147 adNdXi( 0, 6 ) = - c * ( eta + 1.0 );
148 adNdXi( 1, 6 ) = -( ( 2.0 * eta + 1.0 ) * ( xi2 - 1.0 ) ) * 0.5;
149
150 adNdXi( 0, 7 ) = -( ( eta2 - 1.0 ) * ( 2.0 * xi - 1.0 ) ) * 0.5;
151 adNdXi( 1, 7 ) = - c * ( xi - 1.0 );
152
153 adNdXi( 0, 8 ) = 2.0 * xi * ( eta2 - 1.0 );
154 adNdXi( 1, 8 ) = 2.0 * eta * ( xi2 - 1.0 );
155 }
156
157//------------------------------------------------------------------------------
158
159 template<>
160 void
161 InterpolationFunctionTemplate<
162 GeometryType::QUAD, InterpolationType::LAGRANGE, 2, 9 >::d2NdXi2(
163 const Vector< real > & aXi,
164 Matrix< real > & ad2NdXi2 ) const
165 {
166 const real xi = aXi( 0 );
167 const real eta = aXi( 1 );
168
169 // often used constants
170 const real xi2 =xi*xi;
171 const real eta2 = eta*eta;
172
173 ad2NdXi2.set_size( 3, 9 );
174
175 ad2NdXi2( 0, 0 ) = ( eta * ( eta - 1.0 ) ) * 0.5;
176 ad2NdXi2( 1, 0 ) = ( xi * ( xi - 1.0 ) ) * 0.5;
177 ad2NdXi2( 2, 0 ) = ( ( 2.0 * eta - 1.0 ) * ( 2.0 * xi - 1.0 ) ) * 0.25;
178
179 ad2NdXi2( 0, 1 ) = ( eta * ( eta - 1.0 ) ) * 0.5;
180 ad2NdXi2( 1, 1 ) = ( xi * ( xi + 1.0 ) ) * 0.5;
181 ad2NdXi2( 2, 1 ) = ( ( 2.0 * eta - 1.0 ) * ( 2.0 * xi + 1.0 ) ) * 0.25;
182
183 ad2NdXi2( 0, 2 ) = ( eta * ( eta + 1.0 ) ) * 0.5;
184 ad2NdXi2( 1, 2 ) = ( xi * ( xi + 1.0 ) ) * 0.5;
185 ad2NdXi2( 2, 2 ) = ( ( 2.0 * eta + 1.0 ) * ( 2.0 * xi + 1.0 ) ) * 0.25;
186
187 ad2NdXi2( 0, 3 ) = ( eta * ( eta + 1.0 ) ) * 0.5;
188 ad2NdXi2( 1, 3 ) = ( xi * ( xi - 1.0 ) ) * 0.5;
189 ad2NdXi2( 2, 3 ) = ( ( 2.0 * eta + 1.0 ) * ( 2.0 * xi - 1.0 ) ) * 0.25;
190
191 ad2NdXi2( 0, 4 ) = -eta * ( eta - 1.0 );
192 ad2NdXi2( 1, 4 ) = 1.0 - xi2;
193 ad2NdXi2( 2, 4 ) = -xi * ( 2.0 * eta - 1.0 );
194
195 ad2NdXi2( 0, 5 ) = 1.0 - eta2;
196 ad2NdXi2( 1, 5 ) = -xi * ( xi + 1.0 );
197 ad2NdXi2( 2, 5 ) = -eta * ( 2.0 * xi + 1.0 );
198
199 ad2NdXi2( 0, 6 ) = -eta * ( eta + 1.0 );
200 ad2NdXi2( 1, 6 ) = 1.0 - xi2;
201 ad2NdXi2( 2, 6 ) = -xi * ( 2.0 * eta + 1.0 );
202
203 ad2NdXi2( 0, 7 ) = 1.0 - eta2;
204 ad2NdXi2( 1, 7 ) = -xi * ( xi - 1.0 );
205 ad2NdXi2( 2, 7 ) = -eta * ( 2.0 * xi - 1.0 );
206
207 ad2NdXi2( 0, 8 ) = 2.0 * eta2 - 2.0;
208 ad2NdXi2( 1, 8 ) = 2.0 * xi2 - 2.0;
209 ad2NdXi2( 2, 8 ) = 4.0 * eta * xi;
210 }
211
212//------------------------------------------------------------------------------
213 }
214}
215
216#endif //BELFEM_CL_IF_QUAD9_HPP
cl_IF_InterpolationFunctionTemplate.hpp
belfem::Matrix::set_size
void set_size(const size_t aNumRows, const size_t aNumCols)
Definition cl_AR_Matrix.hpp:186
belfem::fem::InterpolationFunctionTemplate
shape function templated class G : Geometry T : Type D : Dimension B : Number of Basis
Definition cl_IF_InterpolationFunctionTemplate.hpp:25
belfem::fem::InterpolationFunctionTemplate::param_coords
void param_coords(Matrix< real > &aXiHat) const override
returns a matrix containing the parameter coordinates of the nodes < number of dimensions x number of...
Definition cl_IF_InterpolationFunctionTemplate.hpp:49
belfem::fem::InterpolationFunctionTemplate::interpolation_order
InterpolationOrder interpolation_order() const override
returns the interpolation order
Definition cl_IF_InterpolationFunctionTemplate.hpp:145
belfem::fem::InterpolationFunctionTemplate::d2NdXi2
void d2NdXi2(const Vector< real > &aXi, Matrix< real > &ad2NdXi2) const override
calculates the second derivative of the shape function in parameter space
Definition cl_IF_InterpolationFunctionTemplate.hpp:110
belfem::fem::InterpolationFunctionTemplate::dNdXi
void dNdXi(const Vector< real > &aXi, Matrix< real > &adNdXi) const override
calculates the first derivative of the shape function in parameter space
Definition cl_IF_InterpolationFunctionTemplate.hpp:89
belfem::fem::InterpolationFunctionTemplate::N
void N(const Vector< real > &aXi, Matrix< real > &aN) const override
evaluates the shape function at a given point
Definition cl_IF_InterpolationFunctionTemplate.hpp:68
belfem::fem
Definition cl_IFB_LINE3.hpp:21
belfem
USER GUIDES:
Definition cl_Capacitor.cpp:16
belfem::InterpolationType::LAGRANGE
@ LAGRANGE
Definition Mesh_Enums.hpp:100
belfem::element_type
ElementType element_type(const std::string &aStr)
Definition Mesh_Enums.hpp:370
belfem::ElementType
ElementType
Element types.
Definition Mesh_Enums.hpp:27
belfem::ElementType::QUAD9
@ QUAD9
Definition Mesh_Enums.hpp:38
belfem::InterpolationOrder
InterpolationOrder
Definition Mesh_Enums.hpp:85
belfem::InterpolationOrder::QUADRATIC
@ QUADRATIC
Definition Mesh_Enums.hpp:88
belfem::real
double real
Definition typedefs.hpp:36
belfem::GeometryType::QUAD
@ QUAD
Definition Mesh_Enums.hpp:74