This term has the same definition as dw_biot_th, but assumes an exponential approximation of the convolution kernel resulting in much higher efficiency. Can use derivatives.
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\begin{array}{l} \int_{\Omega} \left [\int_0^t \alpha_{ij}(t-\tau)\,p(\tau)) \difd{\tau} \right]\,e_{ij}(\ul{v}) \mbox{ ,} \\ \int_{\Omega} \left [\int_0^t \alpha_{ij}(t-\tau) e_{kl}(\ul{u}(\tau)) \difd{\tau} \right] q \end{array}
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dw_biot_eth | (ts, material_0, material_1, virtual, state) |
(ts, material_0, material_1, state, virtual) |
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Evaluate Biot stress tensor.
It is given in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as [11, 22, 33, 12, 13, 23], in 2D it has 3 components with the indices ordered as [11, 22, 12].
Supports ‘eval’, ‘el_avg’ and ‘qp’ evaluation modes.
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- \int_{\Omega} \alpha_{ij} \bar{p}
\mbox{vector for } K \from \Ical_h: - \int_{T_K} \alpha_{ij} \bar{p} / \int_{T_K} 1
- \alpha_{ij} \bar{p}|_{qp}
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ev_biot_stress | (material, parameter) |
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Fading memory Biot term. Can use derivatives.
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\begin{array}{l} \int_{\Omega} \left [\int_0^t \alpha_{ij}(t-\tau)\,p(\tau)) \difd{\tau} \right]\,e_{ij}(\ul{v}) \mbox{ ,} \\ \int_{\Omega} \left [\int_0^t \alpha_{ij}(t-\tau) e_{kl}(\ul{u}(\tau)) \difd{\tau} \right] q \end{array}
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dw_biot_th | (ts, material, virtual, state) |
(ts, material, state, virtual) |
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Biot coupling term with \alpha_{ij} given in vector form exploiting symmetry: in 3D it has the indices ordered as [11, 22, 33, 12, 13, 23], in 2D it has the indices ordered as [11, 22, 12]. Corresponds to weak forms of Biot gradient and divergence terms. Can be evaluated. Can use derivatives.
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\int_{\Omega} p\ \alpha_{ij} e_{ij}(\ul{v}) \mbox{ , } \int_{\Omega} q\ \alpha_{ij} e_{ij}(\ul{u})
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dw_biot | (material, virtual, state) |
(material, state, virtual) | |
(material, parameter_v, parameter_s) |
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