Small deformation elastic contact plane term with penetration penalty.
The plane is given by an anchor point \ul{A} and a normal \ul{n}. The contact occurs in points that orthogonally project onto the plane into a polygon given by orthogonal projections of boundary points \{\ul{B}_i\}, i = 1, \dots, N_B on the plane. In such points, a penetration distance d(\ul{u}) = (\ul{X} + \ul{u} - \ul{A}, \ul{n}) is computed, and a force f(d(\ul{u})) \ul{n} is applied. The force depends on the non-negative parameters k (stiffness) and f_0 (force at zero penetration):
If f_0 = 0:
f(d) = 0 \mbox{ for } d \leq 0 \;, \\ f(d) = k d \mbox{ for } d > 0 \;.
If f_0 > 0:
f(d) = 0 \mbox{ for } d \leq -\frac{2 r_0}{k} \;, \\ f(d) = \frac{k^2}{4 r_0} d^2 + k d + r_0 \mbox{ for } -\frac{2 r_0}{k} < d \leq 0 \;, \\ f(d) = k d + f_0 \mbox{ for } d > 0 \;.
In this case the dependence f(d) is smooth, and a (small) force is applied even for (small) negative penetrations: -\frac{2 r_0}{k} < d \leq 0.
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\int_{\Gamma} \ul{v} \cdot f(d(\ul{u})) \ul{n}
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dw_contact_plane | (material_f, material_n, material_a, material_b, virtual, state) |
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Small deformation elastic contact sphere term with penetration penalty.
The sphere is given by a centre point \ul{C} and a radius R. The contact occurs in points that are closer to \ul{C} than R. In such points, a penetration distance d(\ul{u}) = R - ||\ul{X} + \ul{u} - \ul{C}|| is computed, and a force f(d(\ul{u})) \ul{n}(\ul{u}) is applied, where \ul{n}(\ul{u}) = (\ul{X} + \ul{u} - \ul{C}) / ||\ul{X} + \ul{u} - \ul{C}||. The force depends on the non-negative parameters k (stiffness) and f_0 (force at zero penetration):
If f_0 = 0:
f(d) = 0 \mbox{ for } d \leq 0 \;, \\ f(d) = k d \mbox{ for } d > 0 \;.
If f_0 > 0:
f(d) = 0 \mbox{ for } d \leq -\frac{2 r_0}{k} \;, \\ f(d) = \frac{k^2}{4 r_0} d^2 + k d + r_0 \mbox{ for } -\frac{2 r_0}{k} < d \leq 0 \;, \\ f(d) = k d + f_0 \mbox{ for } d > 0 \;.
In this case the dependence f(d) is smooth, and a (small) force is applied even for (small) negative penetrations: -\frac{2 r_0}{k} < d \leq 0.
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\int_{\Gamma} \ul{v} \cdot f(d(\ul{u})) \ul{n}(\ul{u})
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dw_contact_sphere | (material_f, material_c, material_r, virtual, state) |
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Linear traction forces, where, depending on dimension of ‘material’ argument, \ull{\sigma} \cdot \ul{n} is \bar{p} \ull{I} \cdot \ul{n} for a given scalar pressure, \ul{f} for a traction vector, and itself for a stress tensor.
The material parameter can have one of the following shapes: 1 or (1, 1), (D, 1), (S, 1). The symmetric tensor storage is used in the last case: in 3D S = 6 and the indices ordered as [11, 22, 33, 12, 13, 23], in 2D S = 3 and the indices ordered as [11, 22, 12].
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\int_{\Gamma} \ul{v} \cdot \ull{\sigma} \cdot \ul{n}, \int_{\Gamma} \ul{v} \cdot \ul{n},
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dw_surface_ltr | (opt_material, virtual) |
(opt_material, parameter) |
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Sensitivity of scalar traction.
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\int_{\Gamma} p \nabla \cdot \ul{\Vcal}
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d_sd_surface_integrate | (parameter, parameter_mesh_velocity) |
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“Scalar traction” term, (weak form).
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\int_{\Gamma} q \ul{c} \cdot \ul{n}
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dw_surface_ndot | (material, virtual) |
(material, parameter) |
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Interface jump condition.
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\int_{\Gamma} c\, q (p_1 - p_2)
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dw_jump | (opt_material, virtual, state_1, state_2) |
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