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| 1 | Efficient·C++·finite·element | 1 | Efficient·C++·finite·element |
| 2 | computing·with·Rheolef | 2 | computing·with·Rheolef |
| 3 | Pierre·Saramito | 3 | Pierre·Saramito |
| 4 | version·7.0·update·20·February·2018 | 4 | version·7.0·update·21·February·2018 |
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| 5 | Re·=·10·000 | 5 | Re·=·10·000 |
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| 6 | Bi·=·0.5 | 6 | Bi·=·0.5 |
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| 7 | W·e·=·0.7 | 7 | W·e·=·0.7 |
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| 80 | General·high·order·piecewise·polynomial·finite·element·approximations·are·implemented,·together | 80 | General·high·order·piecewise·polynomial·finite·element·approximations·are·implemented,·together |
| 81 | with·some·mixed·combinations·for·Stokes·and·incompressible·elasticity.·The·characteristic·method | 81 | with·some·mixed·combinations·for·Stokes·and·incompressible·elasticity.·The·characteristic·method |
| 82 | can·be·used·for·diffusion-convection·problems·while·hyperbolic·systems·can·be·discretized·by·the | 82 | can·be·used·for·diffusion-convection·problems·while·hyperbolic·systems·can·be·discretized·by·the |
| 83 | discontinuous·Galerkin·method. | 83 | discontinuous·Galerkin·method. |
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| 84 | \xc4 | 84 | \xc4 |
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| 85 | Rheolef·version·7.0·update·20·February·2018 | 85 | Rheolef·version·7.0·update·21·February·2018 |
| 86 | Contacts | 86 | Contacts |
| 87 | email | 87 | email |
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| 88 | Pierre.Saramito@imag.fr | 88 | Pierre.Saramito@imag.fr |
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| 89 | home·page | 89 | home·page |
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| 350 | . | 350 | . |
| 351 | . | 351 | . |
| 352 | . | 352 | . |
| 353 | . | 353 | . |
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| 354 | \xc6 | 354 | \xc6 |
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| 355 | Rheolef·version·7.0·update·20·February·2018 | 355 | Rheolef·version·7.0·update·21·February·2018 |
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| 356 | 2.4 | 356 | 2.4 |
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| 357 | 2.3.1·The·heat·equation·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·. | 357 | 2.3.1·The·heat·equation·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·. |
| 358 | 2.3.2·The·convection-diffusion·problem·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·. | 358 | 2.3.2·The·convection-diffusion·problem·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·. |
| 359 | The·Navier-Stokes·equations·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·. | 359 | The·Navier-Stokes·equations·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·.·. |
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| 1500 | Index | 1500 | Index |
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| 1501 | 256 | 1501 | 256 |
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| 1502 | \xc8 | 1502 | \xc8 |
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| 1503 | Rheolef·version·7.0·update·20·February·2018 | 1503 | Rheolef·version·7.0·update·21·February·2018 |
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| 1504 | Notations | 1504 | Notations |
| 1505 | Rheolef | 1505 | Rheolef |
| 1506 | d | 1506 | d |
| 1507 | dot(u,v) | 1507 | dot(u,v) |
| 1508 | ddot(sigma,tau) | 1508 | ddot(sigma,tau) |
| 1509 | tr(sigma) | 1509 | tr(sigma) |
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| 1814 | min(phi,psi) | 1814 | min(phi,psi) |
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| 1815 | largest·integral·not·greater·than·φ | 1815 | largest·integral·not·greater·than·φ |
| 1816 | smallest·integral·not·less·than·φ | 1816 | smallest·integral·not·less·than·φ |
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| 1817 | \xc10 | 1817 | \xc10 |
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| 1818 | Rheolef·version·7.0·update·20·February·2018 | 1818 | Rheolef·version·7.0·update·21·February·2018 |
| 1819 | Rheolef | 1819 | Rheolef |
| 1820 | max(phi,psi) | 1820 | max(phi,psi) |
| 1821 | pow(phi,psi) | 1821 | pow(phi,psi) |
| 1822 | atan2(phi,psi) | 1822 | atan2(phi,psi) |
| 1823 | fmod(phi,psi) | 1823 | fmod(phi,psi) |
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| 1824 | mathematics | 1824 | mathematics |
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| 1887 | cd·examples | 1887 | cd·examples |
| 1888 | 11 | 1888 | 11 |
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| 1889 | \xc12 | 1889 | \xc12 |
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| 1890 | 1.1.1 | 1890 | 1.1.1 |
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| 1891 | Rheolef·version·7.0·update·20·February·2018 | 1891 | Rheolef·version·7.0·update·21·February·2018 |
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| 1892 | Problem·statement | 1892 | Problem·statement |
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| 1893 | Let·us·consider·the·classical·Poisson·problem·with·homogeneous·Dirichlet·boundary·conditions·in | 1893 | Let·us·consider·the·classical·Poisson·problem·with·homogeneous·Dirichlet·boundary·conditions·in |
| 1894 | a·domain·bounded·Ω·⊂·Rd·,·d·=·1,·2,·3: | 1894 | a·domain·bounded·Ω·⊂·Rd·,·d·=·1,·2,·3: |
| 1895 | (P):·find·u,·defined·in·Ω,·such·that: | 1895 | (P):·find·u,·defined·in·Ω,·such·that: |
| 1896 | −∆u·= | 1896 | −∆u·= |
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| 2037 | field·uh·(·Xh·); | 2037 | field·uh·(·Xh·); |
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| 2038 | The·field·uh·contains·the·the·degrees·of·freedom. | 2038 | The·field·uh·contains·the·the·degrees·of·freedom. |
| 2039 | uh·[·"·boundary·"·]·=·0; | 2039 | uh·[·"·boundary·"·]·=·0; |
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| 2040 | \xc14 | 2040 | \xc14 |
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| 2041 | Rheolef·version·7.0·update·20·February·2018 | 2041 | Rheolef·version·7.0·update·21·February·2018 |
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| 2042 | Some·degrees·of·freedom·are·prescribed·as·zero·on·the·boundary. | 2042 | Some·degrees·of·freedom·are·prescribed·as·zero·on·the·boundary. |
| 2043 | Let·(ϕi·)06i<dim(Xh·)·be·the·basis·of·Xh·associated·to·the·Lagrange·nodes,·e.g.·the·vertices·of | 2043 | Let·(ϕi·)06i<dim(Xh·)·be·the·basis·of·Xh·associated·to·the·Lagrange·nodes,·e.g.·the·vertices·of |
| 2044 | the·mesh·for·the·P1·approximation·and·the·vertices·and·the·middle·of·the·edges·for·the·P2 | 2044 | the·mesh·for·the·P1·approximation·and·the·vertices·and·the·middle·of·the·edges·for·the·P2 |
| 2045 | approximation.·The·approximate·solution·uh·expresses·as·a·linear·combination·of·the·continuous | 2045 | approximation.·The·approximate·solution·uh·expresses·as·a·linear·combination·of·the·continuous |
| 2046 | piecewise·polynomial·functions·(ϕi·): | 2046 | piecewise·polynomial·functions·(ϕi·): |
| 2047 | X | 2047 | X |
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| 2181 | logo·in·the | 2181 | logo·in·the |
| 2182 | right·margin.·See·http://en.wikipedia.org/wiki/Anaglyph_image·for·more·and·http://www. | 2182 | right·margin.·See·http://en.wikipedia.org/wiki/Anaglyph_image·for·more·and·http://www. |
| 2183 | alpes-stereo.com/lunettes.html·for·how·to·find·anaglyph·red-cyan·glasses.·For·simplicity,·it | 2183 | alpes-stereo.com/lunettes.html·for·how·to·find·anaglyph·red-cyan·glasses.·For·simplicity,·it |
| 2184 | would·perhaps·prefer·to·switch·to·the·gnuplot·render: | 2184 | would·perhaps·prefer·to·switch·to·the·gnuplot·render: |
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| 2185 | \xc16 | 2185 | \xc16 |
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| 2186 | Rheolef·version·7.0·update·20·February·2018 | 2186 | Rheolef·version·7.0·update·21·February·2018 |
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| 2187 | Figure·1.2:·Alternative·representations·of·the·solution·of·the·model·problem·(d·=·2·and·the·P1 | 2187 | Figure·1.2:·Alternative·representations·of·the·solution·of·the·model·problem·(d·=·2·and·the·P1 |
| 2188 | element):·(left)·in·black-and-white;·(right)·in·elevation·and·stereoscopic·anaglyph·mode. | 2188 | element):·(left)·in·black-and-white;·(right)·in·elevation·and·stereoscopic·anaglyph·mode. |
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| 2189 | Figure·1.3:·Red-cyan·anaglyph·glasses·for·the·stereoscopic·visualization. | 2189 | Figure·1.3:·Red-cyan·anaglyph·glasses·for·the·stereoscopic·visualization. |
| 2190 | field·square.field·-gnuplot | 2190 | field·square.field·-gnuplot |
| 2191 | field·square.field·-gnuplot·-bw | 2191 | field·square.field·-gnuplot·-bw |
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| 2239 | Figure·1.4:·Solution·of·the·model·problem·for·d·=·3·and·the·P1·element·:·(left)·mesh;·(right) | 2239 | Figure·1.4:·Solution·of·the·model·problem·for·d·=·3·and·the·P1·element·:·(left)·mesh;·(right) |
| 2240 | isovalue,·cut·planes·and·stereo·anaglyph·renderings. | 2240 | isovalue,·cut·planes·and·stereo·anaglyph·renderings. |
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