Introduction

The solver presented in this article is based on Form-finding with polyhedral meshes made simple by Tang et al.

In the solver, three types of elements can be defined:

• Variables : Values that are optimized by the solver.
• Energies : Objective for the solver whose expression is a linear equation.
• Constraints : Objective for the solver whose expression is a quadratic equation.

Variables

The variables of the Guided Projection Algorithm (GPA) are stored in a column vector, denoted as X in the article. The variables can be geometrically significant (like point coordinates or face normals), meaningful with regard to the constraints (target lengths or forces), or simply dummy elements (as for inequalities).

To facilitate the manipulation and the recognition of the variables, variables are organized into VariableSet. A VariableSet represents a span of the final vector X and contains elements of the same kind with, with the same dimension.

The final vector X corresponds to the successive concatenation of the VariableSet. The order of the VariableSet in X follows that of the declaration.

Optimisation Objectives

The vector X, which contains the variables, can quickly becomes long. Hence expressing the objectives directly on X directly would become cumbersome, especially in terms of index management.

To evade this issue, the objectives (Energy and Constraint) are defined locally on a reduced vector x_red. The x_red vector contains only the necessary variables contained in X. The optimisation goals are first locally expressed using matrices and vectors matching the size of x_red.

During the iteration, the locally defined members based on x_red are completed to be expressed on X.

Energies

In the reference paper, an energy refers to an optimisation objective whose expression is a linear equation. They are organized into two elements:

• EnergyType : Contains the values of k_i and c_i to apply to the local variables in x_red.
• Energy : Contains the list of variables composing x_red and an EnergyType defining the local members for the linear equation .

For more information on energies, refer to the dedicated page.

Constraints

In the reference paper, an constraints refers to an optimisation objective whose expression is a quadratic equation. Two categories of constraints can be defined.

Quadratic Constraint

A quadratic constraint is expressed through a quadratic equation whose members (H_i,red, b_i,red and c_i) are constant. In a similar fashion, quadratic constraint are organized into two elements:

• QuadraticConstraintType : Contains the values of H_i,red, b_i,red and c_i to apply to the local variables in x_red.
• QuadraticConstraint : Contains the list of variables composing x_red and a QuadraticConstraintType defining the local members for the quadratic equation.

Linearised Constraint

A linearised constraint is expressed through a quadratic equation whose members (H_i,red, b_i,red and c_i) depend on the local variables in x_red. Hence the members have to be computed at each iteration. In a similar fashion, linearised constraint are organized into two elements:

• LinearisedConstraintType : Contains the values of H_i,red, b_i,red and c_i to apply to the local variables in x_red and the method to update the members.
• LinearisedConstraint : Contains the list of variables composing x_red and a LinearisedConstraintType defining the local members for the quadratic equation.

For more information on constraints, refer to the dedicated page.

Implementation

Solving Process

Example

For detailed examples, refer to the dedicated page.