 ```python
import numpy as np
import os
from mpi4py import MPI
from tacs import TACS, config, elements, functions, solvers, util
from tacs.exc import TacsException

# Load the structural mesh from a BDF file
bdf_file = 'mesh.bdf'
if not os.path.exists(bdf_file):
    raise TacsException(f'BDF file {bdf_file} does not exist.')

mesh = TACS.get_mesh(bdf_file)

# Set constitutive properties
density = 0.1  # kg/m^3
elastic_modulus = 200e9  # Pa
poisson_ratio = 0.3
shear_correction = 5.0/6.0
yield_stress = 1e8  # Pa
thickness = 0.01  # m

# Loop over components of the mesh, creating stiffness and element objects
for comp in mesh.components:
    el_props = elements.LinearShellProperties(density, elastic_modulus,
                                              poisson_ratio, shear_correction,
                                              yield_stress, thickness)
    comp.setElementProperties(el_props)

# Create a TACS assembler object from the mesh loader
tacs_assembler = TACS.Assembler(mesh)

# Create a KS function and get the design variable values
ks_func = functions.KSTruss()
design_vars = tacs_assembler.getDesignVars()

# Get the node locations and create the forces
node_coords = tacs_assembler.getNodes()
forces = np.zeros(node_coords.shape[0] * 6)

# Set up and solve the analysis problem
displ_vec = util.TACSVector(tacs_assembler)
vel_vec = util.TACSVector(tacs_assembler)
accel_vec = util.TACSVector(tacs_assembler)
resid_vec = util.TACSVector(tacs_assembler)
mass_vec = util.TACSVector(tacs_assembler)

displ_vec.setValues(node_coords)

jac_mat = util.TACSMatrix(tacs_assembler)

tacs_assembler.assemble(displ_vec, resid_vec, jac_mat, ks_func, design_vars,
                        mode=TACS.ASSEMBLY_RESIDUAL)

solver = solvers.Direct()
solver.factor(jac_mat)
solver.solve(resid_vec, displ_vec)

# Evaluate the function and solve for the adjoint variables
ks_func.eval(tacs_assembler, mode=TACS.EVAL_RESIDUAL, design_vars=design_vars,
              displacements=displ_vec, residual=forces)

adj_vec = util.TACSVector(tacs_assembler)
adj_vec.setValues(forces)

tacs_assembler.assemble(displ_vec, resid_vec, jac_mat, ks_func, design_vars,
                        mode=TACS.ASSEMBLY_JACOBIAN, adjoint=adj_vec)

solver.solve(resid_vec, adj_vec)

# Compute the total derivative with respect to material design variables and nodal locations
total_deriv = ks_func.getTotalDeriv(tacs_assembler, design_vars, displ_vec,
                                   adj_vec)

# Create a random direction along which to perturb the nodes and compute the total derivative with respect to nodal locations
np.random.seed(0)
node_perturb = np.random.rand(node_coords.shape[0] * 3).reshape((-1, 3))
node_perturb *= 1e-6

node_coords_perturbed = node_coords.copy()
node_coords_perturbed[:, :2] += node_perturb[:, :2]
node_coords_perturbed[:, 2] = np.maximum(node_coords_perturbed[:, 2], 1e-6)

displ_vec_perturbed = util.TACSVector(tacs_assembler)
displ_vec_perturbed.setValues(node_coords_perturbed)

tacs_assembler.assemble(displ_vec_perturbed, resid_vec, jac_mat, ks_func,
                        design_vars, mode=TACS.ASSEMBLY_RESIDUAL)

resid_vec_perturbed = util.TACSVector(tacs_assembler)
resid_vec_perturbed.setValues(forces)

solver.solve(resid_vec_perturbed, displ_vec_perturbed)

adj_vec_perturbed = util.TACSVector(tacs_assembler)
adj_vec_perturbed.setValues(forces)

tacs_assembler.assemble(displ_vec_perturbed, resid_vec_perturbed, jac_mat,
                        ks_func, design_vars, mode=TACS.ASSEMBLY_JACOBIAN,
                        adjoint=adj_vec_perturbed)

solver.solve(resid_vec_perturbed, adj_vec_perturbed)

total_deriv_nodes = ks_func.getTotalDeriv(tacs_assembler, design_vars,
                                          displ_vec_perturbed, adj_vec_perturbed)

# Set the complex step and compute the perturbed solution
cs_displ_vec = util.TACSVector(tacs_assembler)
cs_displ_vec.setValues(node_coords)

tacs_assembler.setComplexStep(cs_displ_vec)

tacs_assembler.assemble(cs_displ_vec, resid_vec, jac_mat, ks_func, design_vars,
                        mode=TACS.ASSEMBLY_RESIDUAL)

solver.factor(jac_mat)
solver.solve(resid_vec, cs_displ_vec)

cs_forces = np.zeros(forces.shape, dtype=np.complex128)
ks_func.eval(tacs_assembler, mode=TACS.EVAL_RESIDUAL, design_vars=design_vars,
              displacements=cs_displ_vec, residual=cs_forces)

cs_adj_vec = util.TACSVector(tacs_assembler)
cs_adj_vec.setValues(cs_forces)

tacs_assembler.assemble(cs_displ_vec, resid_vec, jac_mat, ks_func, design_vars,
                        mode=TACS.ASSEMBLY_JACOBIAN, adjoint=cs_adj_vec)

solver.solve(resid_vec, cs_adj_vec)

# Evaluate the function for the perturbed solution and compute the projected derivative
cs_total_deriv = ks_func.getTotalDeriv(tacs_assembler, design_vars, cs_displ_vec,
                                       cs_adj_vec)

proj_deriv = np.real(cs_total_deriv - total_deriv)

# Output the results for visualization
output_file = 'output.dat'
with open(output_file, 'w') as f:
    f.write('Total Derivative w.r.t. design variables: {}\n'.format(total_deriv))
    f.write('Total Derivative w.r.t. nodal locations: {}\n'.format(total_deriv_nodes))
    f.write('Projected Derivative w.r.t. nodal locations: {}\n'.format(proj_deriv))
```