Cone constrained Eigenvalue problem subject to external force
$begingroup$
Using FEM I'm trying to model the buckling behaviour of a beam subject to an axial force $P$, a transverse load $q$, and contact boundary conditions, see the figure for a sketch. After discretization etc. I end up with the following system of equations:
begin{align}
(K-lambda I)w &= f \
w & geq 0,
end{align}
with $K$ the stiffness matrix, $lambda$ the critical axial load at which buckling occurs, $I$ the identity, $w$ the transverse displacement, and $f$ the external force vector resulting from the applied load $q$. The constraint $w geq 0$ should be interpreted component-wise, i.e. $w_{i} geq 0, forall i$. I want to solve this problem for $(w,lambda)$. If $f = 0$ we have a cone constrained Eigenvalue problem, and because the cone is the non-negative orthant we even have a Pareto Eigenvalue problem. Such a Pareto Eigenvalue problem can be solved by introducing a complementarity function which guarantees the inequality constrained. However, how can I solve this system for $(lambda,x)$ if $f neq 0$?
optimization eigenvalues-eigenvectors
asked Dec 16 '18 at 11:05
RandyRandy
163
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add a comment |
$begingroup$
Using FEM I'm trying to model the buckling behaviour of a beam subject to an axial force $P$, a transverse load $q$, and contact boundary conditions, see the figure for a sketch. After discretization etc. I end up with the following system of equations:
begin{align}
(K-lambda I)w &= f \
w & geq 0,
end{align}
with $K$ the stiffness matrix, $lambda$ the critical axial load at which buckling occurs, $I$ the identity, $w$ the transverse displacement, and $f$ the external force vector resulting from the applied load $q$. The constraint $w geq 0$ should be interpreted component-wise, i.e. $w_{i} geq 0, forall i$. I want to solve this problem for $(w,lambda)$. If $f = 0$ we have a cone constrained Eigenvalue problem, and because the cone is the non-negative orthant we even have a Pareto Eigenvalue problem. Such a Pareto Eigenvalue problem can be solved by introducing a complementarity function which guarantees the inequality constrained. However, how can I solve this system for $(lambda,x)$ if $f neq 0$?
optimization eigenvalues-eigenvectors
asked Dec 16 '18 at 11:05
RandyRandy
163
$endgroup$
add a comment |
$begingroup$
Using FEM I'm trying to model the buckling behaviour of a beam subject to an axial force $P$, a transverse load $q$, and contact boundary conditions, see the figure for a sketch. After discretization etc. I end up with the following system of equations:
begin{align}
(K-lambda I)w &= f \
w & geq 0,
end{align}
with $K$ the stiffness matrix, $lambda$ the critical axial load at which buckling occurs, $I$ the identity, $w$ the transverse displacement, and $f$ the external force vector resulting from the applied load $q$. The constraint $w geq 0$ should be interpreted component-wise, i.e. $w_{i} geq 0, forall i$. I want to solve this problem for $(w,lambda)$. If $f = 0$ we have a cone constrained Eigenvalue problem, and because the cone is the non-negative orthant we even have a Pareto Eigenvalue problem. Such a Pareto Eigenvalue problem can be solved by introducing a complementarity function which guarantees the inequality constrained. However, how can I solve this system for $(lambda,x)$ if $f neq 0$?
optimization eigenvalues-eigenvectors
asked Dec 16 '18 at 11:05
RandyRandy
163
$endgroup$
Using FEM I'm trying to model the buckling behaviour of a beam subject to an axial force $P$, a transverse load $q$, and contact boundary conditions, see the figure for a sketch. After discretization etc. I end up with the following system of equations:
begin{align}
(K-lambda I)w &= f \
w & geq 0,
end{align}
with $K$ the stiffness matrix, $lambda$ the critical axial load at which buckling occurs, $I$ the identity, $w$ the transverse displacement, and $f$ the external force vector resulting from the applied load $q$. The constraint $w geq 0$ should be interpreted component-wise, i.e. $w_{i} geq 0, forall i$. I want to solve this problem for $(w,lambda)$. If $f = 0$ we have a cone constrained Eigenvalue problem, and because the cone is the non-negative orthant we even have a Pareto Eigenvalue problem. Such a Pareto Eigenvalue problem can be solved by introducing a complementarity function which guarantees the inequality constrained. However, how can I solve this system for $(lambda,x)$ if $f neq 0$?
optimization eigenvalues-eigenvectors
optimization eigenvalues-eigenvectors
asked Dec 16 '18 at 11:05
RandyRandy
163
asked Dec 16 '18 at 11:05
RandyRandy
163
asked Dec 16 '18 at 11:05
RandyRandy
163
asked Dec 16 '18 at 11:05
RandyRandy
163
asked Dec 16 '18 at 11:05
RandyRandy
163
163
add a comment |
add a comment |
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