BELFEM 0.9.0
Berkeley Lab Finite Element Framework
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cl_IF_QUAD8.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_QUAD8_HPP
13#define BELFEM_CL_IF_QUAD8_HPP
14
15
16#include "cl_IF_InterpolationFunctionTemplate.hpp"
17
18namespace belfem
19{
20 namespace fem
21 {
22//------------------------------------------------------------------------------
23
24 template<>
25 InterpolationOrder
26 InterpolationFunctionTemplate<
27 GeometryType::QUAD, InterpolationType::LAGRANGE, 2, 8 >
32
33//------------------------------------------------------------------------------
34
35 template<>
36 ElementType
37 InterpolationFunctionTemplate<
38 GeometryType::QUAD, InterpolationType::LAGRANGE, 2, 8 >
39 ::element_type() const
40 {
41 return ElementType::QUAD8;
42 }
43
44//------------------------------------------------------------------------------
45
46 template<>
47 void
48 InterpolationFunctionTemplate<
49 GeometryType::QUAD, InterpolationType::LAGRANGE, 2, 8 >
50 ::param_coords( Matrix< real > & aXiHat ) const
51 {
52 aXiHat.set_size( 2, 8 );
53
54 aXiHat( 0, 0 ) = -1.000000;
55 aXiHat( 1, 0 ) = -1.000000;
56
57 aXiHat( 0, 1 ) = 1.000000;
58 aXiHat( 1, 1 ) = -1.000000;
59
60 aXiHat( 0, 2 ) = 1.000000;
61 aXiHat( 1, 2 ) = 1.000000;
62
63 aXiHat( 0, 3 ) = -1.000000;
64 aXiHat( 1, 3 ) = 1.000000;
65
66 aXiHat( 0, 4 ) = 0.000000;
67 aXiHat( 1, 4 ) = -1.000000;
68
69 aXiHat( 0, 5 ) = 1.000000;
70 aXiHat( 1, 5 ) = 0.000000;
71
72 aXiHat( 0, 6 ) = 0.000000;
73 aXiHat( 1, 6 ) = 1.000000;
74
75 aXiHat( 0, 7 ) = -1.000000;
76 aXiHat( 1, 7 ) = 0.000000;
77 }
78
79//------------------------------------------------------------------------------
80
81 template<>
82 void
83 InterpolationFunctionTemplate<
84 GeometryType::QUAD, InterpolationType::LAGRANGE, 2, 8 >::N(
85 const Vector< real > & aXi,
86 Matrix< real > & aN ) const
87 {
88 const real xi = aXi( 0 );
89 const real eta = aXi( 1 );
90
91 const real xi2 = xi*xi;
92 const real eta2 = eta*eta;
93
94 aN.set_size( 1, 8 );
95
96 aN( 0, 0 ) = - ( eta - 1.0 ) * ( xi - 1.0 ) * ( eta + xi + 1.0 ) * 0.25;
97 aN( 0, 1 ) = ( eta - 1.0 ) * ( xi + 1.0 ) * ( eta - xi + 1.0 ) * 0.25;
98 aN( 0, 2 ) = ( eta + 1.0 ) * ( xi + 1.0 ) * ( eta + xi - 1.0 ) * 0.25;
99 aN( 0, 3 ) = ( eta + 1.0 ) * ( xi - 1.0 ) * ( - eta + xi + 1.0 ) * 0.25;
100 aN( 0, 4 ) = ( xi2 - 1.0 ) * ( eta - 1.0 ) * 0.5;
101 aN( 0, 5 ) = - ( eta2 - 1.0 ) * ( xi + 1.0 ) * 0.5;
102 aN( 0, 6 ) = - ( xi2 - 1.0 ) * ( eta + 1.0 ) * 0.5;
103 aN( 0, 7 ) = ( eta2 - 1.0 ) * ( xi - 1.0 ) * 0.5;
104 }
105
106//------------------------------------------------------------------------------
107
108 template<>
109 void
110 InterpolationFunctionTemplate<
111 GeometryType::QUAD, InterpolationType::LAGRANGE, 2, 8 >::dNdXi(
112 const Vector< real > & aXi,
113 Matrix< real > & adNdXi ) const
114 {
115 const real xi = aXi( 0 );
116 const real eta = aXi( 1 );
117
118 const real xi2 = xi*xi;
119 const real eta2 = eta*eta;
120
121 adNdXi.set_size( 2, 8 );
122
123 adNdXi( 0, 0 ) = -( eta+xi*2.0 )*( eta-1.0 )*0.25;
124 adNdXi( 1, 0 ) = -( eta*2.0+xi )*( xi-1.0 )*0.25;
125
126 adNdXi( 0, 1 ) = ( eta-xi*2.0 )*( eta-1.0 )*0.25;
127 adNdXi( 1, 1 ) = ( xi+1.0 )*( eta*2.0-xi )*0.25;
128
129 adNdXi( 0, 2 ) = ( eta+xi*2.0 )*( eta+1.0 )*0.25;
130 adNdXi( 1, 2 ) = ( eta*2.0+xi )*( xi+1.0 )*0.25;
131
132 adNdXi( 0, 3 ) = -( eta-xi*2.0 )*( eta+1.0 )*0.25;
133 adNdXi( 1, 3 ) = -( xi-1.0 )*( eta*2.0-xi )*0.25;
134
135 adNdXi( 0, 4 ) = xi*( eta-1.0 );
136 adNdXi( 1, 4 ) = 0.5*xi2-0.5;
137
138 adNdXi( 0, 5 ) = -0.5*eta2+0.5;
139 adNdXi( 1, 5 ) = -eta*( xi+1.0 );
140
141 adNdXi( 0, 6 ) = -xi*( eta+1.0 );
142 adNdXi( 1, 6 ) = -0.5*xi2+0.5;
143
144 adNdXi( 0, 7 ) = 0.5*eta2-0.5;
145 adNdXi( 1, 7 ) = eta*( xi-1.0 );
146 }
147
148//------------------------------------------------------------------------------
149
150 template<>
151 void
152 InterpolationFunctionTemplate<
153 GeometryType::QUAD, InterpolationType::LAGRANGE, 2, 8 >::d2NdXi2(
154 const Vector< real > & aXi,
155 Matrix< real > & ad2NdXi2 ) const
156 {
157 // unpack xi and eta from input vector
158 const real xi = aXi( 0 );
159 const real eta = aXi( 1 );
160
161
162
163 ad2NdXi2.set_size( 3, 8 );
164
165 ad2NdXi2( 0, 0 ) = 0.5 - eta * 0.5;
166 ad2NdXi2( 1, 0 ) = 0.5 - xi * 0.5;
167 ad2NdXi2( 2, 0 ) = 0.25 - 0.5 * ( xi + eta );
168
169 ad2NdXi2( 0, 1 ) = 0.5 - eta * 0.5;
170 ad2NdXi2( 1, 1 ) = xi * 0.5 + 0.5;
171 ad2NdXi2( 2, 1 ) = 0.5 * ( eta - xi ) - 0.25;
172
173 ad2NdXi2( 0, 2 ) = eta * 0.5 + 0.5;
174 ad2NdXi2( 1, 2 ) = xi * 0.5 + 0.5;
175 ad2NdXi2( 2, 2 ) = 0.5 * ( xi + eta ) + 0.25;
176
177 ad2NdXi2( 0, 3 ) = eta * 0.5 + 0.5;
178 ad2NdXi2( 1, 3 ) = 0.5 - xi * 0.5;
179 ad2NdXi2( 2, 3 ) = 0.5 * ( xi - eta ) - 0.25;
180
181 ad2NdXi2( 0, 4 ) = eta - 1.0;
182 ad2NdXi2( 1, 4 ) = 0.0;
183 ad2NdXi2( 2, 4 ) = xi;
184
185 ad2NdXi2( 0, 5 ) = 0.0;
186 ad2NdXi2( 1, 5 ) = - xi - 1.0;
187 ad2NdXi2( 2, 5 ) = - eta;
188
189 ad2NdXi2( 0, 6 ) = - eta - 1.0;
190 ad2NdXi2( 1, 6 ) = 0.0;
191 ad2NdXi2( 2, 6 ) = - xi;
192
193 ad2NdXi2( 0, 7 ) = 0.0;
194 ad2NdXi2( 1, 7 ) = xi - 1.0;
195 ad2NdXi2( 2, 7 ) = eta;
196 }
197
198//------------------------------------------------------------------------------
199 }
200}
201
202#endif //BELFEM_CL_IF_QUAD8_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::QUAD8
@ QUAD8
Definition Mesh_Enums.hpp:44
belfem::InterpolationOrder
InterpolationOrder
Definition Mesh_Enums.hpp:85
belfem::InterpolationOrder::SERENDIPITY
@ SERENDIPITY
Definition Mesh_Enums.hpp:89
belfem::real
double real
Definition typedefs.hpp:36
belfem::GeometryType::QUAD
@ QUAD
Definition Mesh_Enums.hpp:74