To solve the cantilever beam deflection problem using the multifrontal solver in Eigen, you can follow these steps:

1. Set up the finite element model of the beam. This typically involves dividing the beam into a number of elements, and then using the finite element method to approximate the displacement and stress fields within each element. The resulting system of equations can be represented as a linear system of the form Kx = f, where K is the stiffness matrix, x is the vector of unknown nodal displacements, and f is the vector of external loads.
2. Assemble the global stiffness matrix K and load vector f from the element stiffness matrices and load vectors. This involves summing the element stiffness matrices and load vectors over all elements and adding them to the global stiffness matrix and load vector, taking into account the appropriate degrees of freedom (DOFs) at the nodes.
3. Solve the linear system Kx = f using the multifrontal solver in Eigen. This involves calling the lu().solve() function on the stiffness matrix K and passing the load vector f as the argument. The solution x returned by the solver will contain the nodal displacements of the beam.

Here is an example of how you can implement these steps in Eigen:

#include <Eigen/Dense>
#include <iostream>

using namespace Eigen;
using namespace std;

int main() {
  // Set up the finite element model of the beam
  // ...

  // Assemble the global stiffness matrix K and load vector f
  // ...

  // Solve the linear system Kx = f using the multifrontal solver
  VectorXd x = K.lu().solve(f);

  cout << "x = " << endl << x << endl;
  // Output: x = (displacements at each node)

  return 0;
}

To assemble the global stiffness matrix K for a one-dimensional beam using the finite element method, you can follow these steps:

1. Divide the beam into a number of elements of equal length L.
2. For each element, compute the element stiffness matrix k using the beam stiffness formula:

k = (EI / L^3) * [12, 6L, -12, 6L;                  6L, 4L^2, -6L, 2L^2;                  -12, -6L, 12, -6L;                  6L, 2L^2, -6L, 4L^2]

where E is the Young’s modulus of the beam material, I is the second moment of area of the beam cross-section, and L is the element length.

3. For each element, compute the element load vector f by integrating the element load function over the element length:

f = (L / 2) * [f(x1) + f(x2);               (f(x2) - f(x1)) / L]

where x1 and x2 are the coordinates of the end nodes of the element, and f(x) is the element load function.

4. Assemble the global stiffness matrix K and load vector f by looping over all elements and adding the element stiffness matrices and load vectors to the global stiffness matrix and load vector, taking into account the appropriate degrees of freedom (DOFs) at the nodes.

Here is an example of how you can implement these steps in Eigen:

#include <Eigen/Dense>
#include <iostream>

using namespace Eigen;
using namespace std;

int main() {
  // Set up the finite element model of the beam
  int n = 10;  // Number of elements
  double L = 1.0;  // Element length
  double E = 1.0;  // Young's modulus
  double I = 1.0;  // Second moment of area

  // Assemble the global stiffness matrix K and load vector f
  MatrixXd K(2 * n, 2 * n);
  VectorXd f(2 * n);
  for (int i = 0; i < n; i++) {
    // Compute element stiffness matrix k and load vector f
    Matrix2d k;
    Vector2d f;
    k << (E * I / (L * L * L)) * 12, (E * I / (L * L)) * 6 * L,
         (E * I / (L * L)) * 6 * L, (E * I / L) * 4 * L * L;
    f << L / 2 * (f1 + f2), L / 2 * (f2 - f1);

    // Add element stiffness matrix and load vector to global stiffness matrix and load vector
    K.block(2 * i, 2 * i, 2, 2) += k;
    f.segment(2 * i, 2) += f;
  }

  cout << "K = " << endl << K << endl;
  cout << "f = " << endl << f << endl;

  return 0;
}

This code assumes that the element stiffness matrix k and load vector f are already computed for each element, and that the element load function f(x) is given at the end nodes of the element. You can then use the multifrontal solver in Eigen to solve the linear system Kx = f and obtain the nodal displacements of the beam.