One-dimensional modeling of the manufacture of multi-layered material

A computer-readable medium comprising instructions to cause a processor to receive segment data that partitions a process to manufacture a material into a set of segments having one or more layers, wherein the segment data defines layers associated with a component of the manufacturing process and layers associated with the material, and wherein the partitions are defined along a path traversed by the material through the manufacturing process, and further wherein the material has a thickness substantially less than a length and a width of the material;

• receive curvature data for the layers;
• receive initial conditions for a first one of the segments along the path;
• invoke a one-dimensional model based on the initial conditions to calculate values for a process parameter through the layers of a first one of the segments as a function of the curvature data;
• and
• invoke the one-dimensional model for each of the remaining segments using the calculated values for the previous segment along the path from each of the segments as initial conditions for the model.

A computer-readable medium comprising instructions to cause a processor to calculate and display values for a process parameter through a thickness of a manufactured material according to the following equations:

• 24 ∂ ∂ z ⁢ ( k ⁢ ∂ T ∂ z ) j , k = A j ⁢ k j + 1 2 ⁢ T j + 1 - T j z j + 1 - z j - A j - 1 ⁢ k j - 1 2 ⁢ T j - T j - 1 z j - z j - 1 1 2 ⁢ V j + ⁡ ( z j + 1 - z j ) + 1 2 ⁢ V j - ⁡ ( z j - z j - 1 ) ,
• where:
• Aj=1 when a layer is flat, but
• 25 A j = r j + 1 + r j r OD
• when curved,
• Vj+=1 when a layer is flat, but
• 26 V j + = r j + 1 + 3 ⁢ r j 4 ⁢ r OD
• when curved, and
• Vj−=1 when a layer is flat, but
• 27 V j - = r j - 1 + 3 ⁢ r j 4 ⁢ r OD
• when curved.