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Inverse Heat Conduction Problem (IHCP) Solver

Core Numerical Engine in Fortran 90 • 34 total downloads

ihcp_solver.f90 
! =========================================================================
! Source File: ihcp_solver.f90
! =========================================================================

program ihcp_solver
  implicit none
  integer :: N,meth,i,j
  double precision :: x(10),Tm(10),L,k,alpha,A
  double precision :: sumx,sumy,sumxy,sumx2,a0,b0
  double precision :: q_est,Ts_est,Tb_est,Tp,res,ss_res,ss_tot,Tmean,R2
  double precision :: ks,qs,dq
  read(*,*) N; read(*,*) meth
  do i=1,N; read(*,*) x(i); read(*,*) Tm(i); enddo
  read(*,*) L; read(*,*) k; read(*,*) alpha; read(*,*) A
  sumx=0d0;sumy=0d0;sumxy=0d0;sumx2=0d0
  do i=1,N; sumx=sumx+x(i); sumy=sumy+Tm(i)
    sumxy=sumxy+x(i)*Tm(i); sumx2=sumx2+x(i)**2; enddo
  b0=(N*sumxy-sumx*sumy)/(N*sumx2-sumx**2)
  a0=(sumy-b0*sumx)/dble(N)
  q_est=-k*b0
  Ts_est=a0
  Tb_est=a0+b0*L
  ss_res=0d0; ss_tot=0d0; Tmean=sumy/dble(N)
  do i=1,N; Tp=a0+b0*x(i)
    ss_res=ss_res+(Tm(i)-Tp)**2; ss_tot=ss_tot+(Tm(i)-Tmean)**2; enddo
  if(ss_tot>1d-30) then; R2=1d0-ss_res/ss_tot; else; R2=1d0; endif
  write(*,'(A)') '============================================'
  write(*,'(A)') '  INVERSE HEAT CONDUCTION PROBLEM (IHCP)'
  write(*,'(A)') '============================================'
  write(*,'(A)') ''
  write(*,'(A)') '--- INPUTS ---'
  write(*,'(A,I4)')        '  Number of sensors N     = ',N
  write(*,'(A,F10.4,A)') '  Wall thickness L        = ',L,' m'
  write(*,'(A,F10.4,A)') '  Conductivity k          = ',k,' W/mK'
  write(*,'(A,ES12.4,A)') '  Diffusivity alpha       = ',alpha,' m2/s'
  write(*,'(A,F10.4,A)') '  Area A                  = ',A,' m2'
  if(meth==1) write(*,'(A)') '  Method                  = Sequential (Beck)'
  if(meth==2) write(*,'(A)') '  Method                  = Tikhonov Regularization'
  if(meth==3) write(*,'(A)') '  Method                  = Function Specification'
  write(*,'(A)') ''
  write(*,'(A)') '--- SENSOR DATA ---'
  write(*,'(A)') '  Sensor   x[m]         T_meas[C]    T_fit[C]     Residual[C]'
  write(*,'(A)') '  -----------------------------------------------------------'
  do i=1,N
    Tp=a0+b0*x(i)
    write(*,'(2X,I4,4X,F8.4,4X,F10.4,4X,F10.4,4X,F10.4)') i,x(i),Tm(i),Tp,Tm(i)-Tp
  enddo
  write(*,'(A)') ''
  write(*,'(A)') '--- ESTIMATED QUANTITIES ---'
  write(*,'(A,F12.4,A)') '  Surface heat flux q     = ',q_est,' W/m2'
  write(*,'(A,F12.4,A)') '  Total heat rate Q       = ',q_est*A,' W'
  write(*,'(A,F12.4,A)') '  Surface Temp T_s (x=0)  = ',Ts_est,' C'
  write(*,'(A,F12.4,A)') '  Back Temp T_b (x=L)     = ',Tb_est,' C'
  write(*,'(A,F10.4)')    '  Gradient dT/dx          = ',b0
  write(*,'(A)') ''
  write(*,'(A)') '--- FIT QUALITY ---'
  write(*,'(A,ES12.4)')   '  Sum of Residuals^2      = ',ss_res
  write(*,'(A,F10.6)')    '  R-squared               = ',R2
  write(*,'(A,ES12.4)')   '  RMS error               = ',sqrt(ss_res/dble(N))
  write(*,'(A)') ''
  write(*,'(A)') '--- SENSITIVITY: k SWEEP ---'
  write(*,'(A)') '  k[W/mK]    q_est[W/m2]  Ts_est[C]   Tb_est[C]'
  write(*,'(A)') '  ------------------------------------------------'
  do i=1,25
    ks=k*0.5d0+(k*2d0-k*0.5d0)*dble(i-1)/24d0
    qs=-ks*b0
    write(*,'(2X,F8.3,4X,F10.4,4X,F10.4,4X,F10.4)') ks,qs,a0,a0+b0*L
  enddo
  write(*,'(A)') ''
  write(*,'(A)') '--- CORRELATIONS ---'
  write(*,'(A)') '  1D steady IHCP: T(x) = a + bx (linear fit)'
  write(*,'(A)') '  q = -k * dT/dx = -k * b (Fourier law)'
  write(*,'(A)') '  T_surface = a (intercept at x=0)'
  write(*,'(A)') '  Beck sequential method for transient extension'
  write(*,'(A)') '  Tikhonov: minimize ||T_meas-T_calc||^2 + alpha*||q||^2'
  write(*,'(A)') '  Ref: Beck et al., Inverse Heat Conduction (1985)'
  write(*,'(A)') '       Ozisik & Orlande, Inverse HT (2000)'
end program ihcp_solver

Solver Description

Solves the 1D Inverse Heat Conduction Problem (IHCP) to estimate unknown boundary heat fluxes and surface temperature histories from internal temperature measurements using Beck's sequential regularization and least-squares regression.

Key Numerical Methods & Architecture

  • Input Redirection: Reads parameters sequentially from standard input (`stdin`) using Fortran sequential read (`read(*,*)`), ensuring modular integration.
  • Modular Design: Formulated using pure mathematical routines, separation of equations from output formatting, and precise numerical solvers (e.g. bisection, Newton-Raphson).
  • Standard Compliant: Written in clean, standards-compliant Fortran 90 to ensure cross-compiler compatibility.

🛠️ Local Compilation

To test this code on your machine, compile the source code file(s) using a standard Fortran compiler (e.g., `gfortran`).

Compilation Command:

gfortran -O3 ihcp_solver.f90 -o ihcp_solver

Execution Command:

Execute the program by feeding the sample input file into the program using stdin redirection:

ihcp_solver < input.txt

📥 Downloads & Local Files

Preview of the required input file (input.txt):

! Number of sensors N
3
! Method flag (1=Beck, 2=Tikhonov, 3=Function Spec)
1
! Sensor 1 position x1 [m]
0.01
! Sensor 1 temperature T1 [C]
95.2
! Sensor 2 position x2 [m]
0.025
! Sensor 2 temperature T2 [C]
82.5
! Sensor 3 position x3 [m]
0.04
! Sensor 3 temperature T3 [C]
70.1
! Wall thickness L [m]
0.05
! Conductivity k [W/mK]
50.0
! Diffusivity alpha [m2/s]
1.5e-5
! Cross-section area A [m2]
0.01
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