Efficient Model Order Reduction for the Dynamics of Nonlinear Multilayer Sheet Structures with Trial Vector Derivatives

Algorithm 2

Construction of reduction base for jointed structures.

Input: FE-Model, 𝐌, 𝐊

Output: 𝚽=𝚽_𝑆𝚽_𝑉𝚽_𝑇

   %%Computation of trial vectors for the linear system

(1) Compute 𝑠 trial vectors based on static deflection shapes 𝚽_𝑆

(2) Compute 𝑣 trial vectors based on vibration modes 𝚽_𝑉

   %%Computation of trial vector derivatives

(3)  𝐅=[]

(4) for 𝑗=1∶(𝑣+𝑠)

(5)    for 𝑘=−1∶2∶1

(6)        𝜕𝐊/𝜕𝑞_𝑘⋅𝑗 = zeros(𝑛,𝑛)

(7)          for 𝑧=1∶𝑤

(8)           compute 𝑔(𝑧) due to 𝑘𝝋_𝑖

(9)           if (𝑔(𝑧)<=0)

(10)             add 𝑘_𝑛 and 𝑘_𝑡 to 𝜕𝐊/𝜕𝑞_𝑘⋅𝑗 (corresponding to DOF of FE node pair z)

(11)            end

(12)         end

(13)         for 𝑖=1∶(𝑣+𝑠)

(14)          𝐟_{𝑖,𝑘⋅𝑗}=𝜕𝐊𝜕𝑞_𝑘⋅𝑗𝝋_𝑖

(15)          𝐅=𝐅𝐟_{𝑖,𝑘⋅𝑗}

(16)         end

(17)   end

(18) end

(19) use FE software to compute 𝚽_𝑇 based on 𝐊𝚽_𝑇=𝐅

(20) normalize each column of 𝚽_𝑇 with respect to 𝐊

   %%POD

(21) if (POD with inner weighting)

(22)   solve eigenvalue problem 𝚽_𝑇^𝑇𝐊𝚽_𝑇𝝋_𝑇,𝑖=𝜆_𝑖𝝋_𝑇,𝑖

(23) else

(24)   solve eigenvalue problem 𝚽_𝑇^𝑇𝚽_𝑇𝝋_𝑇,𝑖=𝜆_𝑖𝝋_𝑇,𝑖

(25) determine g

(26) 𝚽_𝑇=[]

(27) for 𝑖=1∶𝑔

(28)    𝚽_𝑇=𝚽_𝑇1𝜆_𝑖𝚽_𝑇𝝋_𝑇,𝑖

(29) end