1 Introduction
2 Differential geometry and shell model
2.1 Reference surface
2.2 Geometry of the shell volume
2.3 The 3D shell problem
2.4 Seven-parameter shell model
3 Finite element method
3.1 FEM for shells defined by singular parametrizations
3.2 FEM for implicitly defined shells
4 Numerical results
4.1 Verification example for a parametrically defined shell
4.2 Verification example for an implicitly defined shell
4.3 Pinched hemisphere
4.4 Scordelis-Lo roof
Number of elements | 16 | 64 | 256 |
---|---|---|---|
Linear | 0.0000039 | 0.0000112 | 0.0000373 |
Quadratic | 0.0000215 | 0.0001765 | 0.0026473 |
Cubic | 0.0001254 | 0.0203645 | 0.0823567 |
Quartic | 0.0344561 | 0.0868342 | 0.0921721 |
Quintic | 0.0591080 | 0.0919079 | 0.0924264 |
Sextic | 0.0915553 | 0.0923272 | 0.0924901 |
Septic | 0.0917929 | 0.0924269 | 0.0925234 |
Octic | 0.0923605 | 0.0924707 | 0.0925471 |
Number of elements | 16 | 64 | 256 |
---|---|---|---|
Linear | 1.4e\(+\)05 | 1.5e\(+\)04 | 1.1e\(+\)03 |
Quadratic | 1.1e\(+\)03 | 2.2e\(+\)01 | 3.8e−01 |
Cubic | 1.8e\(+\)01 | 1.5e−01 | 1.0e−03 |
Quartic | 1.7e−01 | 2.2e−04 | 2.4e−07 |
Quintic | 1.0e−03 | 2.7e−07 | 3.9e−08 |
Sextic | 1.1e−05 | 4.7e−07 | 1.8e−07 |
Septic | 8.8e−07 | 6.5e−07 | 8.8e−08 |
Octic | 7.6e−06 | 8.9e−08 | 4.4e−08 |
Number of elements | 4 | 16 | 64 | 256 |
---|---|---|---|---|
Linear | 0.0000011 | 0.0000035 | 0.0000134 | 0.0000526 |
Quadratic | 0.0000062 | 0.0000398 | 0.0005833 | 0.0075576 |
Cubic | 0.0000322 | 0.0011217 | 0.0262481 | 0.0877841 |
Quartic | 0.0002854 | 0.0125512 | 0.0878863 | 0.0923755 |
Quintic | 0.0016276 | 0.0607160 | 0.0922467 | 0.0924516 |
Sextic | 0.0058856 | 0.0894647 | 0.0924056 | 0.0924764 |
Septic | 0.0224888 | 0.0920513 | 0.0924452 | 0.0924957 |
Octic | 0.0572128 | 0.0923485 | 0.0924647 | 0.0925141 |
Number of elements | 4 | 16 | 64 | 256 |
---|---|---|---|---|
Linear | \(-\) 0.0026073 | \(-\) 0.0016144 | \(-\) 0.0044508 | \(-\) 0.0126987 |
Quadratic | \(-\) 0.0019732 | \(-\) 0.0305159 | \(-\) 0.1354229 | \(-\) 0.2741197 |
Cubic | \(-\) 0.0301026 | \(-\) 0.2470338 | \(-\) 0.2968267 | \(-\) 0.3012622 |
Quartic | \(-\) 0.1675085 | \(-\) 0.2967069 | \(-\) 0.3012862 | \(-\) 0.3014015 |
Quintic | \(-\) 0.2888778 | \(-\) 0.3012049 | \(-\) 0.3013835 | \(-\) 0.3014021 |
Sextic | \(-\) 0.2979929 | \(-\) 0.3013161 | \(-\) 0.3014014 | \(-\) 0.3014026 |
Septic | \(-\) 0.3014056 | \(-\) 0.3013603 | \(-\) 0.3014014 | \(-\) 0.3014026 |
Octic | \(-\) 0.3012498 | \(-\) 0.3013926 | \(-\) 0.3014021 | \(-\) 0.3014026 |
4.5 Gyroid
Number of elements | 136 | 544 | 2176 |
---|---|---|---|
Linear | 0.0118106 | 0.0332828 | 0.1055900 |
Quadratic | 0.1694172 | 0.9005433 | 1.7152640 |
Cubic | 1.2274694 | 1.8430397 | 1.8793713 |
Quartic | 1.7745119 | 1.8788403 | 1.8808718 |
Quintic | 1.8726206 | 1.8804255 | 1.8811317 |
Sextic | 1.8784737 | 1.8809134 | 1.8812070 |
Septic | 1.8802114 | 1.8811026 | 1.8812299 |
Octic | 1.8807253 | 1.8811810 | 1.8812370 |