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Loading and Constraints
\beta F = \mu^ {2} C _ {3} + \mu^ {4} C _ {4}.
Skjelbreia and Hendrickson obtain the 18 constants A _ { i j } , \ B _ { i j } , C _ { i } , and c _ { 0 } from matching terms in equal powers of { \bf \dot { \mu } } _ { \mu } and cos \theta in the free surface boundary conditions, Equation 6.2.3-2 and Equation 6.2.3-3. They give the constants as functions of s = \sinh [ \beta ( z _ { s } - z _ { b } ) ] , \ c = \cosh [ \beta ( z _ { s } - z _ { b } ) ] ; as
Loading and Constraints
c _ {0} ^ {2} = g s / c,
A _ {1 1} = 1 / s,
A _ {1 3} = - c ^ {2} \frac {5 c ^ {2} + 1}{8 s ^ {5}},
A _ {1 5} = - \frac {1 1 8 4 c ^ {1} - 1 4 4 0 c ^ {8} - 1 9 9 2 c ^ {6} + 2 6 4 1 c ^ {4} - 2 4 9 c ^ {2} + 1 8}{1 5 3 6 s ^ {1 1}},
A _ {2 2} = \frac {3}{8 s ^ {4}},
A _ {2 4} = \frac {1 9 2 c ^ {8} - 4 2 4 c ^ {6} - 3 1 2 c ^ {4} + 4 8 0 c ^ {2} - 1 7}{7 6 8 s ^ {1 0}},
A _ {3 3} = \frac {1 3 - 4 c ^ {2}}{6 4 s ^ {7}},
A _ {3 5} = \frac {5 1 2 c ^ {1 2} + 4 2 2 4 c ^ {1 0} - 6 8 0 0 c ^ {8} - 1 2 8 0 8 c ^ {6} + 1 6 7 7 0 4 c ^ {4} - 3 1 5 4 c ^ {2} + 1 0 7}{4 0 9 6 s ^ {1 3} (6 c ^ {2} - 1)},
A _ {4 4} = \frac {8 0 c ^ {6} - 8 1 6 c ^ {4} + 1 3 3 8 c ^ {2} - 1 9 7}{1 5 3 6 s ^ {1 0} (6 c ^ {2} - 1)},
A _ {5 5} = - \frac {2 8 8 0 c ^ {1 0} - 7 2 4 8 0 c ^ {8} + 3 2 4 0 0 0 c ^ {6} - 4 3 2 0 0 0 c ^ {4} + 1 6 3 4 7 0 c ^ {2} - 1 6 2 4 5}{6 1 4 4 0 s ^ {1 1} (6 c ^ {2} - 1) (8 c ^ {4} - 1 1 c ^ {2} + 3)},
B _ {2 2} = c \frac {2 c ^ {2} + 1}{4 s ^ {3}},
B _ {2 4} = c \frac {2 7 2 c ^ {8} - 5 0 4 c ^ {6} - 1 9 2 c ^ {4} + 3 2 2 c ^ {2} + 2 1}{3 8 4 s ^ {9}},
B _ {3 3} = 3 \frac {8 c ^ {6} + 1}{6 4 s ^ {6}},
B _ {3 5} = \frac {8 8 1 2 8 c ^ {1 4} - 2 0 8 2 2 4 c ^ {1 2} + 7 0 8 4 8 c ^ {1 0} + 5 4 0 0 0 c ^ {8} - 2 1 8 1 6 c ^ {6} + 6 2 6 4 c ^ {4} - 5 4 c ^ {2} - 8 1}{1 2 2 8 8 s ^ {1 2} (6 c ^ {2} - 1)},
B _ {4 4} = c \frac {7 6 8 c ^ {1 0} - 4 4 8 c ^ {8} - 4 8 c ^ {6} + 4 8 c ^ {4} + 1 0 6 c ^ {2} - 2 1}{3 8 4 s ^ {9} (6 c ^ {2} - 1)},
B _ {5 5} = \frac {1 9 2 0 0 0 c ^ {1 6} - 2 6 2 7 2 0 c ^ {1 4} + 8 3 6 8 0 c ^ {1 2} + 2 0 1 6 0 c ^ {1 0} - 7 2 8 0 c ^ {8}}{1 2 2 8 8 s ^ {1 0} (6 c ^ {2} - 1) (8 c ^ {4} - 1 1 c ^ {2} + 3)}
+ \frac {7 1 6 0 c ^ {6} - 1 8 0 0 c ^ {4} - 1 0 5 0 c ^ {2} + 2 2 5}{1 2 2 8 8 s ^ {1 0} (6 c ^ {2} - 1) (8 c ^ {4} - 1 1 c ^ {2} + 3)},
C _ {1} = \frac {8 c ^ {4} - 8 c ^ {2} + 9}{8 s ^ {4}},
C _ {2} = \frac {3 8 4 0 c ^ {1 2} - 4 0 9 6 c ^ {1 0} - 2 5 9 2 c ^ {8} - 1 0 0 8 c ^ {6} + 5 9 4 4 c ^ {4} - 1 8 3 0 c ^ {2} + 1 4 7}{5 1 2 s ^ {1 0} (6 c ^ {2} - 1)}.
Skjelbreia and Hendrickson (1960) have a factor +2592 multiplying c ^ { 8 } in the equation for C _ { 2 } . This was corrected to -2592 by Nishimura et al. (1970).
They then obtain equations for \beta ( = 2 \pi / \lambda ) and \mu . The wave height is
H = \eta_ {\mathrm{crest}} - \eta_ {\mathrm{trough}} = \eta | _ {\theta = \pi} - \eta | _ {\theta = 0},
so Equation 6.2.3-5 gives
Equation 6.2.3-7
H = \frac {\lambda}{\pi} [ \mu + \mu^ {3} B _ {3 3} + \mu^ {5} (B _ {3 5} + B _ {5 5}) ].
Also, the form assumed for the wave celerity gives
Equation 6.2.3-8
\beta \bar {c} ^ {2} = \frac {2 \pi \lambda}{\tau^ {2}} = c _ {0} ^ {2} (1 + \mu^ {2} C _ {1} + \mu^ {4} C _ {2}).
Given the wave period, wave height, and water depth, Equation 6.2.3-7 and Equation 6.2.3-8 must be solved simultaneously for the wavelength, ¸, and the parameter ¹. This is done with a Newton method, using the Airy (linear) wave solution as an initial guess.
Fluid particle velocities and accelerations for Stokes 5th order wave
The flow potential has been approximated as
\phi = \frac {\lambda \bar {c}}{2 \pi} \sum_ {n = 1} ^ {5} D _ {n} \cosh \frac {2 \pi n}{\lambda} (z - z _ {b}) \sin 2 \pi n \bigg (\frac {S}{\lambda} - \frac {t}{\tau} + \frac {\alpha}{3 6 0} \bigg),
where
D _ {1} = \mu A _ {1 1} + \mu^ {3} A _ {1 3} + \mu^ {5} A _ {1 5},
D _ {2} = - (\mu^ {2} A _ {2 2} + \mu^ {4} A _ {2 4}),
D _ {3} = \mu^ {3} A _ {3 3} + \mu^ {5} A _ {3 5},
D _ {4} = - \mu^ {4} A _ {4 4},
D _ {5} = \mu^ {5} A _ {5 5}.
The fluid particle velocities are
v _ {s} = - \frac {\partial \phi}{\partial s} \quad \text {and} \quad v _ {z} = - \frac {\partial \phi}{\partial z},
and the fluid particle accelerations are
a _ {s} = \frac {\partial v _ {s}}{\partial t} + \frac {\partial v _ {s}}{\partial s} v _ {s} + \frac {\partial v _ {s}}{\partial z} v _ {z} \quad \text {and} \quad a _ {z} = \frac {\partial v _ {z}}{\partial t} + \frac {\partial v _ {z}}{\partial s} v _ {s} + \frac {\partial v _ {z}}{\partial z} v _ {z}.
Loading and Constraints
Since the solution appears in terms of the phase angle µ, it follows that
\frac {\partial \mathbf {v}}{\partial t} = - \bar {c} \frac {\partial \mathbf {v}}{\partial s} = - \frac {\lambda}{\tau} \frac {\partial \mathbf {v}}{\partial s}.
This allows the acceleration components to be written as
a _ {s} = (v _ {s} - \bar {c}) \frac {\partial v _ {s}}{\partial s} + v _ {z} \frac {\partial v _ {s}}{\partial z} \quad \mathrm{and} \quad a _ {z} = (v _ {s} - \bar {c}) \frac {\partial v _ {z}}{\partial s} + v _ {z} \frac {\partial v _ {z}}{\partial z}.
Recall the expression for the dynamic pressure:
p _ {d y n} = \rho \left(\frac {\partial \phi}{\partial t} - \frac {1}{2} \mathbf {v} \cdot \mathbf {v}\right).
Substitution of the expression for Á yields:
v _ {s} = - \bar {c} e _ {1},
v _ {z} = - \bar {c} e _ {2},
a _ {s} = \frac {2 \pi \bar {c}}{\tau} \bigl [ - (1 + e _ {1}) e _ {3} + e _ {2} e _ {4} \bigr ],
a _ {z} = \frac {2 \pi \bar {c}}{\tau} \big [ (1 + e _ {1}) e _ {4} + e _ {2} e _ {3} \big ],
p _ {d y n} = - \rho \left(\frac {\lambda \bar {c}}{\tau} e _ {1} + \frac {1}{2} \mathbf {v} \cdot \mathbf {v}\right),
\frac {\partial p _ {d y n}}{\partial z} = - \frac {\rho 2 \pi \bar {c}}{\tau} \left(e _ {4} + e _ {1} e _ {4} + e _ {2} e _ {3}\right),
\frac {\partial^ {2} p _ {d y n}}{z ^ {2}} = - \frac {\rho 4 \pi^ {2} \bar {c}}{\tau \lambda} \left(e _ {6} + e _ {1} e _ {6} + e _ {4} e _ {4} + e _ {2} e _ {5} + e _ {3} e _ {3}\right),
where
e _ {1} = \sum_ {n = 1} ^ {5} n D _ {n} \cosh \frac {2 \pi n}{\lambda} (z - z _ {b}) \cos 2 \pi n \bigg (\frac {S}{\lambda} - \frac {t}{\tau} + \frac {\alpha}{3 6 0} \bigg),
e _ {2} = \sum_ {n = 1} ^ {5} n D _ {n} \sinh \frac {2 \pi n}{\lambda} (z - z _ {b}) \sin 2 \pi n \bigg (\frac {S}{\lambda} - \frac {t}{\tau} + \frac {\alpha}{3 6 0} \bigg),
e _ {3} = \sum_ {n = 1} ^ {5} n ^ {2} D _ {n} \cosh \frac {2 \pi n}{\lambda} (z - z _ {b}) \sin 2 \pi n \bigg (\frac {S}{\lambda} - \frac {t}{\tau} + \frac {\alpha}{3 6 0} \bigg),
e _ {4} = \sum_ {n = 1} ^ {5} n ^ {2} D _ {n} \sinh \frac {2 \pi n}{\lambda} (z - z _ {b}) \cos 2 \pi n \bigg (\frac {S}{\lambda} - \frac {t}{\tau} + \frac {\alpha}{3 6 0} \bigg),
e _ {5} = \sum_ {n = 1} ^ {5} n ^ {3} D _ {n} \sinh \frac {2 \pi n}{\lambda} (z - z _ {b}) \sin 2 \pi n \bigg (\frac {S}{\lambda} - \frac {t}{\tau} + \frac {\alpha}{3 6 0} \bigg),
and
e _ {6} = \sum_ {n = 1} ^ {5} n ^ {3} D _ {n} \cosh \frac {2 \pi n}{\lambda} (z - z _ {b}) \cos 2 \pi n \left(\frac {S}{\lambda} - \frac {t}{\tau} + \frac {\alpha}{3 6 0}\right).
Finally, the surface position is given as
\eta = \frac {\lambda}{2 \pi} \sum_ {n = 1} ^ {5} E _ {n} \cos 2 \pi n \bigg (\frac {S}{\lambda} - \frac {t}{\tau} + \frac {\alpha}{3 6 0} \bigg),
where
E _ {1} = - \mu ,
E _ {2} = \mu^ {2} B _ {2 2} + \mu^ {4} B _ {2 4},
E _ {3} = - (\mu^ {3} B _ {3 3} + \mu^ {5} B _ {3 5}),
E _ {4} = \mu^ {4} B _ {4 4},
E _ {5} = - \mu^ {5} B _ {5 5}.
The Stokes wave field is a spatial description of the wave field. All wave field quantities are calculated up to the instantaneous fluid level. The wave field defines velocity, acceleration, and dynamic pressure at spatial locations for all values of time. Hence, the velocity, acceleration, and dynamic pressure are determined by using the current (for geometrically nonlinear analysis) or reference (for geometrically linear analysis) location of the structure at the current time in the appropriate equations. The time used in the wave field equations is the total time for the analysis, which accumulates over all steps in the analysis (*STATIC, *DYNAMIC, etc.).
6.3 Pressure penetration loading
6.3.1 Pressure penetration loading with surface-based contact
ABAQUS/Standard allows for the simulation of fluid penetrating into the surface between two
contacting bodies and application of the fluid pressure normal to the surfaces. The capability is invoked by using the *PRESSURE PENETRATION option, as described in ``Pressure penetration loading,'' Section 21.3.5 of the ABAQUS/Standard User's Manual. The surface-based contact approach is used to model the interactions between the bodies, where one surface definition provides the "master" surface and the other surface definition provides the "slave" surface. Both surfaces can be deformable, or one can be rigid.
A single slave node-based penetration criterion is used. Fluid will penetrate into the surface between the contacting bodies from one or multiple locations, which are exposed to the fluid, until a point is reached where the contact pressure is greater than the critical value specified on the *PRESSURE PENETRATION option, cutting off further penetration of the fluid. The critical contact pressure is introduced to account for the asperities on the contacting surfaces. The higher this value, the easier the fluid penetrates. The default value of the critical contact pressure is zero, in which case fluid penetration occurs only if the contact pressure is zero and contact is lost. When a node has a contact status of "OPEN," its nearest neighboring nodes are considered to be subjected to the fluid pressure as well. The nodes initially exposed to the fluid, which are specified on the *PRESSURE
PENETRATION option, will always be subjected to the fluid pressure irrespective of the contact status at these nodes.
The pressure penetration load will be applied normal to the element surface based on the pressure penetration criterion described above at the beginning of an increment and will remain constant over that increment even if the fluid penetrates further during that increment. A nodal integration scheme is used to integrate the distributed pressure penetration load over an element; the variation of the distributed load over an element will be determined by the load magnitudes at the element's nodes, which are coincident with the base points. Consider the contact interaction of three nodes--101, 102, and 103--on the slave surface made up of faces of two first-order elements, 1 and 2, with a master surface made up of faces of two elements, 4 and 5, which are described by nodes 201, 202, and 203 as shown in Figure 6.3.1-1. If the fluid with a pressure magnitude of f has penetrated up to node 102 on the slave surface, the variation of the distributed load over element 1 is given by
P _ {1} = f N _ {1} + f N _ {2} = f,
and the variation of the distributed load over element 2 is given by
P _ {2} = f N _ {1},
where N _ { 1 } and N _ { 2 } are the shape functions on the face of a first-order element.
For a deformable master surface we must also consider how the fluid pressure is applied to the master surface, which depends on the location of the "anchor" point on the master surface. The "anchor" point is chosen as the point on the master surface closest to the last slave node subjected to the fluid pressure (pressure penetration tip). All the nodes between the "anchor point" and the node initially exposed to the fluid on the master surface, as specified on the *PRESSURE PENETRATION option, are considered to be subjected to the fluid pressure.
Figure 6.3.1-1 Pressure penetration with nonmatching meshes.
flowchart
graph TD
A["1"] -->|fluid flow direction| B["201"]
A -->|fluid flow direction| C["202"]
A -->|fluid flow direction| D["D"]
A -->|fluid flow direction| E["203"]
A -->|fluid flow direction| F["101"]
A -->|fluid flow direction| G["102"]
A -->|fluid flow direction| H["103"]
A -->|fluid flow direction| I["4"]
A -->|fluid flow direction| J["5"]
A -->|fluid flow direction| K["master surface"]
The pressure penetration capability does not require that the contacting surfaces have matching meshes; however, the best accuracy is obtained when meshes are initially matching. For initially nonmatching meshes the equivalent fluid pressure applied on the master surface can be evaluated based on equilibrium considerations. For the problem illustrated in Figure 6.3.1-1, the "anchor" point corresponding to slave node 102 is D. For the element associated with the anchor point (element 5) on the master surface, part of the load is transferred from element 1 and part of the load is transferred from element 2 on the slave surface. The load distribution on element 5 is illustrated in Figure 6.3.1-2, where L _ { 5 } is the element length. Because the fluid pressure will be simulated as a distributed load on the contacting surfaces, the variation of the load over an element needs to be described. For a first-order element if Q _ { 2 0 2 } and Q _ { 2 0 3 } are the equivalent forces at nodes 202 and 203 due to the distributed load shown in Figure 6.3.1-2, the variation of the distributed load over element 5 on the master surface is given by
P _ {5} = \frac {4 Q _ {2 0 2} - 2 Q _ {2 0 3}}{L _ {5}} N _ {1} + \frac {4 Q _ {2 0 3} - 2 Q _ {2 0 2}}{L _ {5}} N _ {2}.
A similar approach is used for the second-order elements.
Figure 6.3.1-2 Stress distribution over an element on the master surface with nonmatching meshes.
text_image
P₁ P₂ 202 D (102) (103) 203 L₅ master surface
6.4 Multi-point constraints
6.4.1 Sliding constraint
Loading and Constraints
The sliding constraint has a variety of uses. For example, this MPC is used in conjunction with other MPC types to constrain a shell element mesh to a solid element mesh. The MPC maintains consistency with standard shell theory by forcing initially straight lines through the thickness to remain straight despite rotation and displacement. When applied to solid element nodes on the shell-solid interface, this MPC enforces a kinematic approximation of compatibility with the shell model.
The theory of this constraint is as follows:
Let P ^ { 1 } , P ^ { n } be the points defining the line; and let P ^ { m } be the node that must lie on this line. The direction of the line is given by
\mathbf {n} = (\mathbf {x} ^ {n} - \mathbf {x} ^ {1}) / l,
where
l = \left[ \left(\mathbf {x} ^ {n} - \mathbf {x} ^ {1}\right) \cdot \left(\mathbf {x} ^ {n} - \mathbf {x} ^ {1}\right) \right] ^ {\frac {1}{2}}.
Let i; j; k be base vectors in the x \dash , y - , z-directions in the global coordinate system. Then, define a unit vector normal to the line as
\mathbf {t} ^ {1} = \frac {\mathbf {n} \times \mathbf {i}}{| \mathbf {n} \times \mathbf {i} |}
unless { \bf n } = { \bf i } ( n _ { 2 } = n _ { 3 } = 0 ) , in which case we use
\mathbf {t} ^ {1} = \frac {\mathbf {n} \times \mathbf {k}}{| \mathbf {n} \times \mathbf {k} |}.
Now we can define an orthogonal normal as
\mathbf {t} ^ {2} = \mathbf {n} \times \mathbf {t} ^ {1}.
\mathbf { t } ^ { 1 } ; \mathbf { t } ^ { 2 } , and n now form a set of orthonormal base vectors with \mathbf { t } ^ { 1 } and \mathbf { t } ^ { 2 } normal to the line joining P ^ { 1 } and P ^ { n } . The constraint can be imposed by the condition that the line joining the node m to node 1 be perpendicular to \mathbf { t } ^ { 1 } and \mathbf { t } ^ { 2 } . ; that is,
\left(\mathbf {x} ^ {m} - \mathbf {x} ^ {1}\right) \cdot \mathbf {t} ^ {1} = 0
and
\left(\mathbf {x} ^ {m} - \mathbf {x} ^ {1}\right) \cdot \mathbf {t} ^ {2} = 0.
We now choose a local coordinate numbering system such that i is the global direction on which \mathbf { t } ^ { 1 } has its largest projection:
Loading and Constraints
\left| t _ {i} ^ {1} \right| > \left| t _ {j} ^ {1} \right| \text { and } \left| t _ {i} ^ {1} \right| \geq \left| t _ {k} ^ {1} \right|, \text { where } j \neq i, k \neq i.
Likewise, we choose global direction j such that j 6= i and
| t _ {j} ^ {2} | > | t _ {k} ^ {2} |, \text {where} j \neq k.
Using this definition of i; j; k the constraint conditions can be written explicitly in terms of coordinate components of node m as
x _ {i} ^ {m} t _ {i} ^ {1} + x _ {j} ^ {m} t _ {j} ^ {1} + x _ {k} ^ {m} t _ {k} ^ {1} = \mathbf {x} ^ {1} \cdot \mathbf {t} ^ {1}
and
x _ {i} ^ {m} t _ {i} ^ {2} + x _ {j} ^ {m} t _ {j} ^ {2} + x _ {k} ^ {m} t _ {k} ^ {2} = \mathbf {x} ^ {1} \cdot \mathbf {t} ^ {2}.
These equations can be used to eliminate xmi ; xmj (note that the numbering of i; j; k avoids dividing through by zero in this elimination):
x _ {i} ^ {m} [ t _ {i} ^ {1} t _ {j} ^ {2} - t _ {i} ^ {2} t _ {j} ^ {1} ] = \mathbf {x} \cdot (\mathbf {t} ^ {1} t _ {j} ^ {2} - \mathbf {t} ^ {2} t _ {j} ^ {1}) - x _ {k} ^ {m} (t _ {k} ^ {1} t _ {j} ^ {2} - t _ {k} ^ {2} t _ {j} ^ {1})
and
x _ {j} ^ {m} [ t _ {i} ^ {1} t _ {j} ^ {2} - t _ {j} ^ {1} t _ {i} ^ {2} ] = \mathbf {x} \cdot (\mathbf {t} ^ {2} t _ {i} ^ {1} - \mathbf {t} ^ {1} t _ {j} ^ {1}) - x _ {k} ^ {m} (t _ {k} ^ {1} t _ {i} ^ {2} - t _ {k} ^ {2} t _ {i} ^ {1}).
The above equations will enforce the desired constraint. We also need the derivatives of these constraints. These are
\left(d \mathbf {x} ^ {m} - d \mathbf {x} ^ {1}\right) \cdot \mathbf {t} ^ {1} + \left(\mathbf {x} ^ {m} - \mathbf {x} ^ {1}\right) \cdot d \mathbf {t} ^ {1} = 0
and
\left(d \mathbf {x} ^ {m} - d \mathbf {x} ^ {1}\right) \cdot \mathbf {t} ^ {2} + \left(\mathbf {x} ^ {m} - \mathbf {x} ^ {1}\right) \cdot d \mathbf {t} ^ {2} = 0,
where
d \mathbf {t} ^ {1} = (- \mathbf {t} ^ {1} \cdot d \mathbf {n}) \mathbf {n}
and
d \mathbf {t} ^ {2} = (- \mathbf {t} ^ {2} \cdot d \mathbf {n}) \mathbf {n}.
These equations reduce to
Loading and Constraints
\left(d \mathbf {x} ^ {m} - d \mathbf {x} ^ {1}\right) \cdot \mathbf {t} ^ {1} - \left(\mathbf {x} ^ {m} - \mathbf {x} ^ {1}\right) \cdot \mathbf {n t} ^ {1} \cdot d \mathbf {n} = 0
and
\left(d \mathbf {x} ^ {m} - d \mathbf {x} ^ {1}\right) \cdot \mathbf {t} ^ {2} - \left(\mathbf {x} ^ {m} - \mathbf {x} ^ {1}\right) \cdot \mathbf {n t} ^ {2} \cdot d \mathbf {n} = 0.
dn can be obtained from the definition of n to give
d \mathbf {n} = \frac {1}{l} (\mathbf {i} - \mathbf {n n}) \cdot (d \mathbf {x} ^ {n} - d \mathbf {x} ^ {1}),
and, therefore,
\mathbf {t} ^ {1} \cdot d \mathbf {n} = \left(\mathbf {t} ^ {1} / l\right) \cdot \left(d \mathbf {x} ^ {n} - d \mathbf {x} ^ {1}\right)
and
\mathbf {t} ^ {2} \cdot d \mathbf {n} = \left(\mathbf {t} ^ {2} / l\right) \cdot \left(d \mathbf {x} ^ {n} - d \mathbf {x} ^ {1}\right).
The incremental constraint equations become
d \mathbf {x} ^ {m} \cdot \mathbf {t} ^ {1} - d \mathbf {x} ^ {1} \cdot \mathbf {t} ^ {1} - \left(\mathbf {x} ^ {m} - \mathbf {x} ^ {1}\right) \cdot \mathbf {n} \left(\mathbf {t} ^ {1} / l\right) \cdot \left(d \mathbf {x} ^ {n} - d \mathbf {x} ^ {1}\right) = 0
and
d \mathbf {x} ^ {m} \cdot \mathbf {t} ^ {2} - d \mathbf {x} ^ {1} \cdot \mathbf {t} ^ {1} - (\mathbf {x} ^ {m} - \mathbf {x} ^ {1}) \cdot \mathbf {n} (\mathbf {t} ^ {2} / l) \cdot (d \mathbf {x} ^ {n} - d \mathbf {x} ^ {1}) = 0.
Let P = 1 / l ( \mathbf { x } ^ { m } - \mathbf { x } ^ { 1 } ) \cdot \mathbf { n } . Then, the above equations, when written out in full with the same ordering of i; j; k used above, are
d x _ {i} ^ {m} t _ {i} ^ {1} + d x _ {j} ^ {m} t _ {j} ^ {1} + d x _ {k} ^ {m} t _ {k} ^ {1} - P d \mathbf {x} ^ {n} \cdot \mathbf {t} ^ {1} - (1 - P) d \mathbf {x} ^ {1} \cdot \mathbf {t} ^ {1} = 0
and
d x _ {i} ^ {m} t _ {i} ^ {2} + d x _ {j} ^ {m} t _ {j} ^ {2} + d x _ {k} ^ {m} t _ {k} ^ {2} - P d \mathbf {x} ^ {n} \cdot \mathbf {t} ^ {2} - (1 - P) d \mathbf {x} ^ {1} \cdot \mathbf {t} ^ {2} = 0.
Solving for dxmi ; dxmj we obtain
\begin{array}{l} d x _ {i} ^ {m} (t _ {i} ^ {1} t _ {j} ^ {2} - t _ {i} ^ {2} t _ {j} ^ {1}) + d x _ {k} ^ {m} (t _ {k} ^ {1} t _ {j} ^ {2} - t _ {k} ^ {2} t _ {j} ^ {1}) + \\ - P d \mathbf {x} ^ {n} \cdot (\mathbf {t} ^ {1} t _ {j} ^ {2} - \mathbf {t} ^ {2} t _ {j} ^ {1}) - (1 - P) d \mathbf {x} ^ {1} \cdot (\mathbf {t} ^ {1} t _ {j} ^ {2} - \mathbf {t} ^ {2} t _ {j} ^ {1}) = 0 \\ \end{array}
and