Initial value problems

The finite element (FE) information and methods are defined in a Matlab object called NL_2D_FEM. An instance of the FE model (i.e. a new FE model) can be created using the default constructor: model = NL_2D_FEM. This creates an object model containing the data of a given problem and methods to perform the data analysis.

The attributes and methods of the NL_2D_FEM class (denoted model) are given below:

Attributes

1. model.geom: The geometry of the model (key points and lines of the geometry)

   • model.geom.geom_node: List of number and coordinates of the key points of the geometry. Each line is of the type (point number, x coordinate, y coordinate).

   • model.geom.geom_element: List of lines in the geometry connecting the nodes. Each line is of the type (line number, point 1, point 2).

   • model.geom.discretisation: List of number of elements in each line (list of integers, with size the number of lines of the geometry).

2. model.prop: Element cross-section and material properties (constant properties for all elements of the model)

   • model.prop.area: Area of the cross-section.

   • model.prop.inertia: Second moment of area of the cross-section.

   • model.prop.shear_coeff_k: Shear coefficient of the cross-section.

   • model.prop.density: Material density.

   • model.prop.young_mod: Material Young’s modulus.

   • model.prop.shear_mod: Material shear modulus.

   • model.prop.poisson: Material Poisson’s ratio.

   • model.prop.alpha: Material damping coefficient (linear Rayleigh damping).

3. model.mesh: mesh of the FE model

   • model.mesh.nodes: List of number and coordinates of the nodes of the mesh. Each line is of the type (node number, x coordinate, y coordinate).

   • model.mesh.connect: Connectivity table, textit{i.e.} list of elements of the mesh. Each line is of the type (element number, node 1, node 2).

   • model.mesh.number_nodes: Total number of nodes in the mesh.

   • model.mesh.number_elements: Total number of elements in the mesh.

4. model.visu: Visualization information for displaying results

   • model.visu.visu_node_list: List (cell array) of observed node and degree of freedom (dof).

     ** model.visu.visu_node_list{n} is a structure with two fields: a) mds.visu.visu_node_list{n}.node: Node number, and b) mds.visu.visu_node_list{n}.dof: List of dof number(s).

   • model.visu.dof: List of indices of observed dof in the assembled vector.

5. model.boundary: Boundary conditions information

   • model.boundary.bc_node_list: List (cell array) of boundary condition nodes and dof.

     ** mds.boundary.bc_node_list{n} is a structure with two fields: a) mds.boundary.bc_node{n}.node: Node number, and b) mds.boundary.bc_node{n}.dof: List of dof number(s).

   • model.boundary.dof_list: List of indices of all dof in the assembled vector.

   • model.boundary.prescribed_dof: List of indices of all prescribed dof (displacement or rotation set to zero).

   • model.boundary.active_dof: List of indices of all non-boundary condition dof.

6. model.loads: Loads (static and periodic, point force and distributed loads)

model.loads.static: Static loads.

• model.loads.static.static_ponctual_force_node_list: List (cell array) of forced node(s) and dof.

  ** model.loads.static.static_ponctual_force_node_list{n} is a structure with three fields: a) model.loads.static.static_ponctual_force_node_list{n}.node: Node number, b) model.loads.static.static_ponctual_force_node_list{n}.dof: List of dof number(s), and c) model.loads.static.static_ponctual_force_node_list{n}.amplitude: List of force amplitudes (one amplitude for each dof in the previous field).

• model.loads.static.static_distributed_force_direction: List of directions among (1) x-axis (2) y-axis or (3) moment.

• model.loads.static.static_distributed_force_amplitude: List of amplitudes (linear force density).

model.loads.periodic: Periodic loads

• model.loads.periodic.periodic_ponctual_force_node_list: List (cell array) of forced node(s) and dof.

  ** model.loads.periodic.periodic_ponctual_force_node_list{n} is a structure with three fields: a) model.loads.periodic.periodicponctual_force_node_list{n}.node: Node number, b) model.loads.periodic.periodic_ponctual_force_node_list{n}.dof: List of dof number(s), and c) model.loads.periodic.periodic_ponctual_force_node_list{n}.amplitude: List of force amplitudes (one amplitude for each dof in the previous field).

• model.loads.periodic.periodic_distributed_force_direction: List of direction among (1) x-axis (2) y-axis or (3) moment.

• model.loads.periodic.periodic_distributed_force_amplitude: List of amplitudes (linear force density).

7. model.matrices: Assembled matrices of the model

   • model.matrices.mass: Mass matrix.

   • model.matrices.damping: Damping matrix.

   • model.matrices.stiffness_at_origin: Linear stiffness matrix (around undeformed (reference) configuration).

   • model.matrices.tangent_stiffness_at_qs: Unused.

8. model.vectors: Assembled vectors of the model

   • model.vectors.null_vector: Assembled vector of dof full of zeros.

   • model.vectors.static_forces: Assembled vector of static forces.

   • model.vectors.periodic_forces: Assembled vector (matrix) of periodic forces (each row corresponds to a harmonic).

   • model.vectors.static_solution: Unused.

9. model.solver: Solver information

   • model.solver.type: 'homemade': Default, solve NL system with a simple Newton method, or 'fsolve': solve NL system with Matlab fsolve from optimization toolbox).

   • model.solver.tol_res: Residual tolerance.

   • model.solver.tol_step: Step tolerance.

   • model.solver.n_iter_max: Number of maximum iterations for Newton’s method.

Methods

Initialization methods

1. model.set_model_from_mds: Set the model attributes from a model data structure mds.

2. model.set_geom: Set the geometry of the model (key points and lines of the geometry).

3. model.set_prop: Set the cross-section and material properties.

4. model.set_boundary: Set the boundary conditions.

5. model.set_visu: Set the visualization information for displaying result.

6. model.set_static_loads: Set the static loads (point force and distributed loads).

7. model.set_periodic_loads: Set the periodic loads (point force and distributed loads).

8. model.set_mesh: Set mesh from user inputs.

FEM methods

Mesh-related methods

1. model.auto_mesh: Create a mesh from the geometry information.

2. model.mesh_from_truss: Used in auto_mesh.

Element-related methods

1. model.elementOrientation: Compute element length and orientation angle.

2. model.transformationMatrix: Compute rotation matrix from an orientation angle.

3. model.elementary_mass_matrix: Elementary mass matrix of the Timoshenko beam model.

4. model.elementary_tangent_stiffness_matrix: Elementary tangent stiffness matrix from a given (elementary) dof vector.

5. model.elementary_internal_force_vector: Elementary vector of nonlinear internal forces from a given (elementary) dof vector.

Assembly-related methods

1. model.assemble_constant_matrix: Assemble a matrix depending on the input (textit{e.g.} mass, stiffness).

2. model.assemble_constant_vector: Assemble a vector depending on the input (textit{e.g.} [static, periodic, internal] forces).

3. model.initialise_matrices_and_vector: Assemble the mass and stiffness matrices and the static and periodic force vectors.

Mechanics-related methods

1. model.strains_and_stress_at_gauss_point: Return stains, stress and gradient information from a given dof vector.

2. model.solve_static_problem: Solve the static problem.

3. model.linear_modal_analysis: Linear modal analysis around a given configuration.

4. model.buckling_analysis: Simple buckling analysis.

5. model.linear_analysis: Linear forced response.

MAN-related methods

1. model.set_MAN_parameters: Set the parameters structure.

2. model.initialise_MAN_computation: Automatic initialization of a MAN computation. Performs a static, modal anlysis around a static configuration and computes a MAN starting point (U0) depending on the type of computation.

3. model.solve_MAN_system_at_fixed_amplitude: Solve the MAN system such that the first harmonic of a given dof is fixed (variable frequency); used in the initialization method.

4. model.solve_MAN_system_at_fixed_frequency: Solve the MAN system such that the frequency is fixed (variable amplitude); used in the initialization method.

5. model.man_initial_point: Returns a MAN point from a given static and periodic vector of dof.

6. model.man_auxiliary_variable_vector: Computes the MAN auxiliary variables vector from a given vector of dof.

7. model.auxiliary_variable_expression: Used in the MAN equation to define the auxiliary variables.

Equation (residual)-related methods

1. model.static_residual: Used to solve the static problem.

2. model.newton_solver: Homemade Newton solver used to solve the static problem or the MAN initial point (or use the Matlab function fsolve from the optimization toolbox).

Display-related methods

1. model.plot_deformed_mesh: Plot deformed structure from a given vector.

2. model.plot_static_configuration: Plot the static configuration.

3. model.plot_mode_shape: Plot the mode shapes from a given list.