BELFEM 0.9.0
Berkeley Lab Finite Element Framework
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cl_IF_HEX8.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_HEX8_HPP
13#define BELFEM_CL_IF_HEX8_HPP
14
15#include "cl_IF_InterpolationFunctionTemplate.hpp"
16
17namespace belfem
18{
19 namespace fem
20 {
21//------------------------------------------------------------------------------
22
23 template<>
24 inline InterpolationOrder
25 InterpolationFunctionTemplate<
26 GeometryType::HEX, InterpolationType::LAGRANGE, 3, 8 >
31
32//------------------------------------------------------------------------------
33
34 template<>
35 inline ElementType
36 InterpolationFunctionTemplate<
37 GeometryType::HEX, InterpolationType::LAGRANGE, 3, 8 >
38 ::element_type() const
39 {
40 return ElementType::HEX8;
41 }
42
43//------------------------------------------------------------------------------
44
45 template<>
46 inline void
47 InterpolationFunctionTemplate<
48 GeometryType::HEX, InterpolationType::LAGRANGE, 3, 8 >
49 ::param_coords( Matrix< real > & aXiHat ) const
50 {
51 aXiHat.set_size( 3, 8 );
52
53 aXiHat( 0, 0 ) = -1.000000;
54 aXiHat( 1, 0 ) = -1.000000;
55 aXiHat( 2, 0 ) = -1.000000;
56
57 aXiHat( 0, 1 ) = 1.000000;
58 aXiHat( 1, 1 ) = -1.000000;
59 aXiHat( 2, 1 ) = -1.000000;
60
61 aXiHat( 0, 2 ) = 1.000000;
62 aXiHat( 1, 2 ) = 1.000000;
63 aXiHat( 2, 2 ) = -1.000000;
64
65 aXiHat( 0, 3 ) = -1.000000;
66 aXiHat( 1, 3 ) = 1.000000;
67 aXiHat( 2, 3 ) = -1.000000;
68
69 aXiHat( 0, 4 ) = -1.000000;
70 aXiHat( 1, 4 ) = -1.000000;
71 aXiHat( 2, 4 ) = 1.000000;
72
73 aXiHat( 0, 5 ) = 1.000000;
74 aXiHat( 1, 5 ) = -1.000000;
75 aXiHat( 2, 5 ) = 1.000000;
76
77 aXiHat( 0, 6 ) = 1.000000;
78 aXiHat( 1, 6 ) = 1.000000;
79 aXiHat( 2, 6 ) = 1.000000;
80
81 aXiHat( 0, 7 ) = -1.000000;
82 aXiHat( 1, 7 ) = 1.000000;
83 aXiHat( 2, 7 ) = 1.000000;
84 }
85
86//------------------------------------------------------------------------------
87
88 template<>
89 inline void
90 InterpolationFunctionTemplate<
91 GeometryType::HEX, InterpolationType::LAGRANGE, 3, 8 >::N(
92 const Vector< real > & aXi,
93 Matrix< real > & aN ) const
94 {
95 const real xi = aXi( 0 );
96 const real eta = aXi( 1 );
97 const real zeta = aXi( 2 );
98
99 aN.set_size( 1, 8 );
100
101 aN( 0, 0 ) = - ( eta - 1.0 ) * ( xi - 1.0 ) * ( zeta - 1.0 ) * 0.125;
102 aN( 0, 1 ) = ( eta - 1.0 ) * ( xi + 1.0 ) * ( zeta - 1.0 ) * 0.125;
103 aN( 0, 2 ) = - ( eta + 1.0 ) * ( xi + 1.0 ) * ( zeta - 1.0 ) * 0.125;
104 aN( 0, 3 ) = ( eta + 1.0 ) * ( xi - 1.0 ) * ( zeta - 1.0 ) * 0.125;
105 aN( 0, 4 ) = ( eta - 1.0 ) * ( xi - 1.0 ) * ( zeta + 1.0 ) * 0.125;
106 aN( 0, 5 ) = - ( eta - 1.0 ) * ( xi + 1.0 ) * ( zeta + 1.0 ) * 0.125;
107 aN( 0, 6 ) = ( eta + 1.0 ) * ( xi + 1.0 ) * ( zeta + 1.0 ) * 0.125;
108 aN( 0, 7 ) = - ( eta + 1.0 ) * ( xi - 1.0 ) * ( zeta + 1.0 ) * 0.125;
109 }
110
111//------------------------------------------------------------------------------
112
113 template<>
114 inline void
115 InterpolationFunctionTemplate<
116 GeometryType::HEX, InterpolationType::LAGRANGE, 3, 8 >::dNdXi(
117 const Vector< real > & aXi,
118 Matrix< real > & adNdXi ) const
119 {
120 const real xi = aXi( 0 );
121 const real eta = aXi( 1 );
122 const real zeta = aXi( 2 );
123
124 adNdXi.set_size( 3, 8 );
125
126 adNdXi( 0, 0 ) = -( eta - 1 ) * ( zeta - 1 ) * 0.125;
127 adNdXi( 1, 0 ) = -( xi - 1 ) * ( zeta - 1 ) * 0.125;
128 adNdXi( 2, 0 ) = -( eta - 1 ) * ( xi - 1 ) * 0.125;
129
130 adNdXi( 0, 1 ) = ( eta - 1 ) * ( zeta - 1 ) * 0.125;
131 adNdXi( 1, 1 ) = ( xi + 1 ) * ( zeta - 1 ) * 0.125;
132 adNdXi( 2, 1 ) = ( eta - 1 ) * ( xi + 1 ) * 0.125;
133
134 adNdXi( 0, 2 ) = -( eta + 1 ) * ( zeta - 1 ) * 0.125;
135 adNdXi( 1, 2 ) = -( xi + 1 ) * ( zeta - 1 ) * 0.125;
136 adNdXi( 2, 2 ) = -( eta + 1 ) * ( xi + 1 ) * 0.125;
137
138 adNdXi( 0, 3 ) = ( eta + 1 ) * ( zeta - 1 ) * 0.125;
139 adNdXi( 1, 3 ) = ( xi - 1 ) * ( zeta - 1 ) * 0.125;
140 adNdXi( 2, 3 ) = ( eta + 1 ) * ( xi - 1 ) * 0.125;
141
142 adNdXi( 0, 4 ) = ( eta - 1 ) * ( zeta + 1 ) * 0.125;
143 adNdXi( 1, 4 ) = ( xi - 1 ) * ( zeta + 1 ) * 0.125;
144 adNdXi( 2, 4 ) = ( eta - 1 ) * ( xi - 1 ) * 0.125;
145
146 adNdXi( 0, 5 ) = -( eta - 1 ) * ( zeta + 1 ) * 0.125;
147 adNdXi( 1, 5 ) = -( xi + 1 ) * ( zeta + 1 ) * 0.125;
148 adNdXi( 2, 5 ) = -( eta - 1 ) * ( xi + 1 ) * 0.125;
149
150 adNdXi( 0, 6 ) = ( eta + 1 ) * ( zeta + 1 ) * 0.125;
151 adNdXi( 1, 6 ) = ( xi + 1 ) * ( zeta + 1 ) * 0.125;
152 adNdXi( 2, 6 ) = ( eta + 1 ) * ( xi + 1 ) * 0.125;
153
154 adNdXi( 0, 7 ) = -( eta + 1 ) * ( zeta + 1 ) * 0.125;
155 adNdXi( 1, 7 ) = -( xi - 1 ) * ( zeta + 1 ) * 0.125;
156 adNdXi( 2, 7 ) = -( eta + 1 ) * ( xi - 1 ) * 0.125;
157 }
158
159//------------------------------------------------------------------------------
160
161 template<>
162 inline void
163 InterpolationFunctionTemplate<
164 GeometryType::HEX, InterpolationType::LAGRANGE, 3, 8 >::d2NdXi2(
165 const Vector< real > & aXi,
166 Matrix< real > & ad2NdXi2 ) const
167 {
168 real xi = aXi( 0 );
169 real eta = aXi( 1 );
170 real zeta = aXi( 2 );
171
172 ad2NdXi2.set_size( 6, 8 );
173
174 ad2NdXi2( 0, 0 ) = 0.0;
175 ad2NdXi2( 1, 0 ) = 0.0;
176 ad2NdXi2( 2, 0 ) = 0.0;
177 ad2NdXi2( 3, 0 ) = 0.125 * ( - xi + 1.0 );
178 ad2NdXi2( 4, 0 ) = 0.125 * ( - eta + 1.0 );
179 ad2NdXi2( 5, 0 ) = 0.125 * ( - zeta + 1.0 );
180
181 ad2NdXi2( 0, 1 ) = 0.0;
182 ad2NdXi2( 1, 1 ) = 0.0;
183 ad2NdXi2( 2, 1 ) = 0.0;
184 ad2NdXi2( 3, 1 ) = 0.125 * ( xi + 1.0 );
185 ad2NdXi2( 4, 1 ) = 0.125 * ( eta - 1.0 );
186 ad2NdXi2( 5, 1 ) = 0.125 * ( zeta - 1.0 );
187
188 ad2NdXi2( 0, 2 ) = 0.0;
189 ad2NdXi2( 1, 2 ) = 0.0;
190 ad2NdXi2( 2, 2 ) = 0.0;
191 ad2NdXi2( 3, 2 ) = 0.125 * ( - xi - 1.0 );
192 ad2NdXi2( 4, 2 ) = 0.125 * ( - eta - 1.0 );
193 ad2NdXi2( 5, 2 ) = 0.125 * ( - zeta + 1.0 );
194
195 ad2NdXi2( 0, 3 ) = 0.0;
196 ad2NdXi2( 1, 3 ) = 0.0;
197 ad2NdXi2( 2, 3 ) = 0.0;
198 ad2NdXi2( 3, 3 ) = 0.125 * ( xi - 1.0 );
199 ad2NdXi2( 4, 3 ) = 0.125 * ( eta + 1.0 );
200 ad2NdXi2( 5, 3 ) = 0.125 * ( zeta - 1.0 );
201
202 ad2NdXi2( 0, 4 ) = 0.0;
203 ad2NdXi2( 1, 4 ) = 0.0;
204 ad2NdXi2( 2, 4 ) = 0.0;
205 ad2NdXi2( 3, 4 ) = 0.125 * ( xi - 1.0 );
206 ad2NdXi2( 4, 4 ) = 0.125 * ( eta - 1.0 );
207 ad2NdXi2( 5, 4 ) = 0.125 * ( zeta + 1.0 );
208
209 ad2NdXi2( 0, 5 ) = 0.0;
210 ad2NdXi2( 1, 5 ) = 0.0;
211 ad2NdXi2( 2, 5 ) = 0.0;
212 ad2NdXi2( 3, 5 ) = 0.125 * ( - xi - 1.0 );
213 ad2NdXi2( 4, 5 ) = 0.125 * ( - eta + 1.0 );
214 ad2NdXi2( 5, 5 ) = 0.125 * ( - zeta - 1.0 );
215
216 ad2NdXi2( 0, 6 ) = 0.0;
217 ad2NdXi2( 1, 6 ) = 0.0;
218 ad2NdXi2( 2, 6 ) = 0.0;
219 ad2NdXi2( 3, 6 ) = 0.125 * ( xi + 1.0 );
220 ad2NdXi2( 4, 6 ) = 0.125 * ( eta + 1.0 );
221 ad2NdXi2( 5, 6 ) = 0.125 * ( zeta + 1.0 );
222
223 ad2NdXi2( 0, 7 ) = 0.0;
224 ad2NdXi2( 1, 7 ) = 0.0;
225 ad2NdXi2( 2, 7 ) = 0.0;
226 ad2NdXi2( 3, 7 ) = 0.125 * ( - xi + 1.0 );
227 ad2NdXi2( 4, 7 ) = 0.125 * ( - eta - 1.0 );
228 ad2NdXi2( 5, 7 ) = 0.125 * ( - zeta - 1.0 );
229 }
230
231//------------------------------------------------------------------------------
232 }
233}
234
235#endif //BELFEM_CL_IF_HEX8_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::HEX8
@ HEX8
Definition Mesh_Enums.hpp:33
belfem::InterpolationOrder
InterpolationOrder
Definition Mesh_Enums.hpp:85
belfem::InterpolationOrder::LINEAR
@ LINEAR
Definition Mesh_Enums.hpp:87
belfem::real
double real
Definition typedefs.hpp:36
belfem::GeometryType::HEX
@ HEX
Definition Mesh_Enums.hpp:76