BELFEM 0.9.0
Berkeley Lab Finite Element Framework
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cl_IF_PLATE.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_PLATE_HPP
13#define BELFEM_CL_IF_PLATE_HPP
14
15#include "cl_IF_InterpolationFunctionTemplate.hpp"
16namespace belfem
17{
18 namespace fem
19 {
20//------------------------------------------------------------------------------
21
22 template<>
23 InterpolationOrder
24 InterpolationFunctionTemplate<
25 GeometryType::QUAD, InterpolationType::HERMITE, 2, 16 >
30
31//------------------------------------------------------------------------------
32
33 template<>
34 ElementType
35 InterpolationFunctionTemplate<
36 GeometryType::QUAD, InterpolationType::HERMITE, 2, 16 >
37 ::element_type() const
38 {
39 return ElementType::QUAD4;
40 }
41
42//------------------------------------------------------------------------------
43
44 template<>
45 void
46 InterpolationFunctionTemplate<
47 GeometryType::QUAD, InterpolationType::HERMITE, 2, 16 >
48 ::param_coords( Matrix< real > & aXiHat ) const
49 {
50 aXiHat.set_size( 2, 4 );
51
52 aXiHat( 0, 0 ) = -1.000000;
53 aXiHat( 1, 0 ) = -1.000000;
54
55 aXiHat( 0, 1 ) = 1.000000;
56 aXiHat( 1, 1 ) = -1.000000;
57
58 aXiHat( 0, 2 ) = 1.000000;
59 aXiHat( 1, 2 ) = 1.000000;
60
61 aXiHat( 0, 3 ) = -1.000000;
62 aXiHat( 1, 3 ) = 1.000000;
63 }
64
65//------------------------------------------------------------------------------
66
67 template<>
68 void
69 InterpolationFunctionTemplate<
70 GeometryType::QUAD, InterpolationType::HERMITE, 2, 16 >::N(
71 const Vector< real > & aXi,
72 Matrix< real > & aN ) const
73 {
74 const real xi = aXi( 0 );
75 const real eta = aXi( 1 );
76
77 const real a = xi-1.0;
78 const real b = xi+1.0;
79 const real c = eta-1.0;
80 const real d = eta+1.0;
81 const real e = xi-2.0;
82 const real f = xi+2.0;
83 const real g = eta-2.0;
84 const real h = eta+2.0;
85 const real a2 = a*a;
86 const real b2 = b*b;
87 const real c2 = c*c;
88 const real d2 = d*d;
89 aN.set_size( 1, 16 );
90
91 aN( 0, 0 ) = c2*h*a2*f;
92 aN( 0, 1 ) = c2*h*a2*b;
93 aN( 0, 2 ) = c2*d*a2*f;
94 aN( 0, 3 ) = c2*d*a2*b;
95 aN( 0, 4 ) = -c2*h*b2*e;
96 aN( 0, 5 ) = c2*h*a*b2;
97 aN( 0, 6 ) = -c2*d*b2*e;
98 aN( 0, 7 ) = c2*d*a*b2;
99 aN( 0, 8 ) = d2*g*b2*e;
100 aN( 0, 9 ) = -d2*g*a*b2;
101 aN( 0, 10 ) = -c*d2*b2*e;
102 aN( 0, 11 ) = c*d2*a*b2;
103 aN( 0, 12 ) = -d2*g*a2*f;
104 aN( 0, 13 ) = -d2*g*a2*b;
105 aN( 0, 14 ) = c*d2*a2*f;
106 aN( 0, 15 ) = c*d2*a2*b;
107
108 aN *= 0.0625;
109 }
110
111//------------------------------------------------------------------------------
112
113 template<>
114 void
115 InterpolationFunctionTemplate<
116 GeometryType::QUAD, InterpolationType::HERMITE, 2, 16 >::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
123 const real a = xi-1.0;
124 const real b = xi+1.0;
125 const real c = eta-1.0;
126 const real d = eta+1.0;
127 const real e = xi-2.0;
128 const real f = xi+2.0;
129 const real g = eta-2.0;
130 const real h = eta+2.0;
131 const real p = a*b;
132 const real q = c*d;
133 const real r = 1.0+xi*(2.0-3.0*xi);
134 const real s = (3.0*xi+2.0)*xi-1.0;
135 const real u = 1.0 + eta*( 2.0-3.0*eta);
136 const real v = (3.0*eta+2.0)*eta - 1.0;
137 const real a2 = a*a;
138 const real b2 = b*b;
139 const real c2 = c*c;
140 const real d2 = d*d;
141
142 adNdXi.set_size( 2, 16 );
143
144 adNdXi( 0, 0 ) = 3.0*p*c2*h;
145 adNdXi( 1, 0 ) = 3.0*q*a2*f;
146
147 adNdXi( 0, 1 ) = -c2*h*r;
148 adNdXi( 1, 1 ) = 3.0*q*a2*b;
149
150 adNdXi( 0, 2 ) = 3.0*p*c2*d;
151 adNdXi( 1, 2 ) = -a2*f*u;
152
153 adNdXi( 0, 3 ) = -c2*d*r;
154 adNdXi( 1, 3 ) = -a2*b*u;
155
156 adNdXi( 0, 4 ) = -3.0*p*c2*h;
157 adNdXi( 1, 4 ) = -3.0*q*b2*e;
158
159 adNdXi( 0, 5 ) = c2*h*s;
160 adNdXi( 1, 5 ) = 3.0*q*a*b2;
161
162 adNdXi( 0, 6 ) = -3.0*p*c2*d;
163 adNdXi( 1, 6 ) = b2*e*u;
164
165 adNdXi( 0, 7 ) = c2*d*s;
166 adNdXi( 1, 7 ) = -a*b2*u;
167
168 adNdXi( 0, 8 ) = 3.0*p*d2*g;
169 adNdXi( 1, 8 ) = 3.0*q*b2*e;
170
171 adNdXi( 0, 9 ) = -d2*g*s;
172 adNdXi( 1, 9 ) = -3.0*q*a*b2;
173
174 adNdXi( 0, 10 ) = -3.0*p*c*d2;
175 adNdXi( 1, 10 ) = -b2*e*v;
176
177 adNdXi( 0, 11 ) = c*d2*s;
178 adNdXi( 1, 11 ) = a*b2*v;
179
180 adNdXi( 0, 12 ) = -3.0*p*d2*g;
181 adNdXi( 1, 12 ) = -3.0*q*a2*f;
182
183 adNdXi( 0, 13 ) = d2*g*r;
184 adNdXi( 1, 13 ) = -3.0*q*a2*b;
185
186 adNdXi( 0, 14 ) = 3.0*p*c*d2;
187 adNdXi( 1, 14 ) = a2*f*v;
188
189 adNdXi( 0, 15 ) = -c*d2*r;
190 adNdXi( 1, 15 ) = a2*b*v;
191
192 adNdXi *= 0.0625;
193 }
194
195//------------------------------------------------------------------------------
196
197 template<>
198 void
199 InterpolationFunctionTemplate<
200 GeometryType::QUAD, InterpolationType::HERMITE, 2, 16 >::d2NdXi2(
201 const Vector< real > & aXi,
202 Matrix< real > & ad2NdXi2 ) const
203 {
204 const real xi = aXi( 0 );
205 const real eta = aXi( 1 );
206
207 const real a = xi-1.0;
208 const real b = xi+1.0;
209 const real c = eta-1.0;
210 const real d = eta+1.0;
211 const real e = xi-2.0;
212 const real f = xi+2.0;
213 const real g = eta-2.0;
214 const real h = eta+2.0;
215 const real p = a*b;
216 const real q = c*d;
217 const real r = 1.0+xi*(2.0-3.0*xi);
218 const real s = (3.0*xi+2.0)*xi-1.0;
219 const real u = 1.0 + eta*( 2.0-3.0*eta);
220 const real v = (3.0*eta+2.0)*eta - 1.0;
221
222 const real w = (3.0*xi-1.0);
223 const real x = (3.0*xi+1.0);
224 const real y = (3.0*eta-1.0);
225 const real z = (3.0*eta+1.0);
226
227 const real a2 = a*a;
228 const real b2 = b*b;
229 const real c2 = c*c;
230 const real d2 = d*d;
231
232 ad2NdXi2.set_size( 3, 16 );
233
234 ad2NdXi2( 0, 0 ) = 6.0*xi*c2*h;
235 ad2NdXi2( 1, 0 ) = 6.0*eta*a2*f;
236 ad2NdXi2( 2, 0 ) = 9.0*q*p;
237
238 ad2NdXi2( 0, 1 ) = 2.0*w*c2*h;
239 ad2NdXi2( 1, 1 ) = 6.0*eta*a2*b;
240 ad2NdXi2( 2, 1 ) = -3.0*q*r;
241
242 ad2NdXi2( 0, 2 ) = 6.0*xi*c2*d;
243 ad2NdXi2( 1, 2 ) = 2.0*y*a2*f;
244 ad2NdXi2( 2, 2 ) = -3.0*p*u;
245
246 ad2NdXi2( 0, 3 ) = 2.0*w*c2*d;
247 ad2NdXi2( 1, 3 ) = 2.0*y*a2*b;
248 ad2NdXi2( 2, 3 ) = u*r;
249
250 ad2NdXi2( 0, 4 ) = -6.0*xi*c2*h;
251 ad2NdXi2( 1, 4 ) = -6.0*eta*b2*e;
252 ad2NdXi2( 2, 4 ) = -9.0*q*p;
253
254 ad2NdXi2( 0, 5 ) = 2.0*x*c2*h;
255 ad2NdXi2( 1, 5 ) = 6.0*eta*a*b2;
256 ad2NdXi2( 2, 5 ) = 3.0*q*s;
257
258 ad2NdXi2( 0, 6 ) = -6.0*xi*c2*d;
259 ad2NdXi2( 1, 6 ) = -2.0*y*b2*e;
260 ad2NdXi2( 2, 6 ) = 3.0*p*u;
261
262 ad2NdXi2( 0, 7 ) = 2.0*x*c2*d;
263 ad2NdXi2( 1, 7 ) = 2.0*y*a*b2;
264 ad2NdXi2( 2, 7 ) = -u*s;
265
266 ad2NdXi2( 0, 8 ) = 6.0*xi*d2*g;
267 ad2NdXi2( 1, 8 ) = 6.0*eta*b2*e;
268 ad2NdXi2( 2, 8 ) = 9.0*q*p;
269
270 ad2NdXi2( 0, 9 ) = -2.0*x*d2*g;
271 ad2NdXi2( 1, 9 ) = -6.0*eta*a*b2;
272 ad2NdXi2( 2, 9 ) = -3.0*q*s;
273
274 ad2NdXi2( 0, 10 ) = -6.0*xi*c*d2;
275 ad2NdXi2( 1, 10 ) = -2.0*z*b2*e;
276 ad2NdXi2( 2, 10 ) = -3.0*p*v;
277
278 ad2NdXi2( 0, 11 ) = 2.0*x*c*d2;
279 ad2NdXi2( 1, 11 ) = 2.0*z*a*b2;
280 ad2NdXi2( 2, 11 ) = v*s;
281
282 ad2NdXi2( 0, 12 ) = -6.0*xi*d2*g;
283 ad2NdXi2( 1, 12 ) = -6.0*eta*a2*f;
284 ad2NdXi2( 2, 12 ) = -9.0*q*p;
285
286 ad2NdXi2( 0, 13 ) = -2.0*w*d2*g;
287 ad2NdXi2( 1, 13 ) = -6.0*eta*a2*b;
288 ad2NdXi2( 2, 13 ) = 3.0*q*r;
289
290 ad2NdXi2( 0, 14 ) = 6.0*xi*c*d2;
291 ad2NdXi2( 1, 14 ) = 2.0*z*a2*f;
292 ad2NdXi2( 2, 14 ) = 3.0*p*v;
293
294 ad2NdXi2( 0, 15 ) = 2.0*w*c*d2;
295 ad2NdXi2( 1, 15 ) = 2.0*z*a2*b;
296 ad2NdXi2( 2, 15 ) = -v*r;
297
298 ad2NdXi2*= 0.0625;
299 }
300
301//------------------------------------------------------------------------------
302 }
303}
304#endif //BELFEM_CL_IF_PLATE_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::HERMITE
@ HERMITE
Definition Mesh_Enums.hpp:101
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::QUAD4
@ QUAD4
Definition Mesh_Enums.hpp:31
belfem::MaterialProperty::q
@ q
Definition cl_Material.hpp:176
belfem::InterpolationOrder
InterpolationOrder
Definition Mesh_Enums.hpp:85
belfem::InterpolationOrder::CUBIC
@ CUBIC
Definition Mesh_Enums.hpp:90
belfem::real
double real
Definition typedefs.hpp:36
belfem::GeometryType::QUAD
@ QUAD
Definition Mesh_Enums.hpp:74