Verification problem: Maxwell damping in 3DEC

Note

To view this project in 3DEC, use the menu command Help ► Examples…. The main data files used are shown at the end of this example. The remaining data files can be found in the project.

This verification problem considers a 100-m long column. Horizontal velocities are initialized with a sine wave. Several tests are run with different stiffness properties to simulate different frequencies of vibration. Maxwell damping is applied and the damping versus frequency is compared to the analytical solution.

Rigid Blocks

In the first test, the column is composed of 100 rigid blocks and the joint stiffnesses are altered to generate diffrent frequencies of the system. The damping versus frequency in the model compared to the analytical solution is shown in Figure 1.

../../../../../../_images/rigid.png

Figure 1: Response of rigid block model run for 21 different frequencies.

Deformable Block

The same model is run with a single deformable block and the zone moduli are modified to produce different frequencies. The results are shown in Figure 2.

../../../../../../_images/zones.png

Figure 2: Response of zoned block model run for 21 different frequencies.

Deformable Blocks with Joints

In this example, 100 deformable blocks form the column and both joint and zone stiffnesses are altered to geenrate different frequency responses. The results are shown in Figure 3.

../../../../../../_images/zones-joints.png

Figure 3: Response of jointed and zoned block model run for 21 different frequencies.

Data Files

benchmark-rigid.dat

; Benchmark test forigin Maxwell Damping

; model new
model title 'Maxwell Damping: Vibration of 1D column'
model configure dynamic
model large-strain off
fish automatic-create off
;
fish define Set_Parameters
    ; ------- input parameters ------------------    
    ;global target_frequency = 1  ; frequency in hz - set in master file
    global num_cycles = 3
    
    global column_height = 100.0
    global height_zones = 100
        
    global soil_density = 2.0;
    local  wave_length = 4.0*column_height
    local  shear_vel = wave_length*target_frequency  
    local  period = 1. / target_frequency
    global num_periods = num_cycles*period
    global soil_shear = shear_vel*shear_vel*soil_density
    global soil_poisson = 0.3
    global soil_bulk = 2.*soil_shear*(1.+soil_poisson)
    soil_bulk = soil_bulk / (3.0*(1.0-2.0*soil_poisson))

    global bheight = 1.0
    global jks = soil_shear * bheight
    global jkn = jks * 2.0*(1.0+soil_poisson)

end
[Set_Parameters]
;
block create brick 0 1 0 1 0 [column_height]
block cut joint-set dip 0 dip-direction 0 spacing [bheight] origin 0 0 0
block create brick 0 1 0 1 -1 0

block property density [soil_density]

block insitu stress -1e-6 -1e-6 -1e-6 0 0 0 

block contact jmodel assign mohr
block contact property stiffness-normal [jkn] stiffness-shear [jks] ...
  friction 45 cohesion 1e20 tension 1e20

; --- Maxwell Damping Properties forigin 5% Damping ------------
block mechanical damping maxwell (0.0385 0.5) (0.0335 3.5) (0.052 25.0)
;
block fix range position-z -1 0
block fix velocity-y velocity-z rotation range position-z 0 [column_height]


;
; ---- initialize velocit to quarter wave length mode ---
fish define initial_velocity
    local max_vel = 1.0   
;    loop foreach local p_gp block.gp.list
;        local zcord = block.gp.pos.z(p_gp)
;        local theta = (math.pi/2.)*zcord/column_height
;        block.gp.vel.x(p_gp) = max_vel*math.sin(theta)
;    end_loop
    loop foreach local p_b block.list
        local zcord = block.pos.z(p_b)
        if zcord > 0
          local theta = (math.pi/2.)*zcord/column_height
          block.vel.x(p_b) = max_vel*math.sin(theta)
        endif
    end_loop
end
[initial_velocity]
;
model history dynamic time-total
block history displacement-x position 0 0 [column_height]
block history velocity-x position 0 0 [column_height]
block contact history stress-shear-x position 0 0 1

model solve time-total [num_periods]
;
history export '3' vs '1' table '10'
;
program call 'process.fis'
[data_process('rigid.csv')]

benchmark-zones.dat

; Benchmark test forigin Maxwell Damping

; model new
model title 'Maxwell Damping: Vibration of 1D column'
model configure dynamic
model large-strain off
fish automatic-create off
;
fish define Set_Parameters
    ; ------- input parameters ------------------    
    ; global target_frequency = 1  ; frequency in hz
    global num_cycles = 3
    
    global column_height = 100.0
    global height_zones = 100
        
    global soil_density = 2.0;
    local  wave_length = 4.0*column_height
    local  shear_vel = wave_length*target_frequency  
    local  period = 1. / target_frequency
    global num_periods = num_cycles*period
    global soil_shear = shear_vel*shear_vel*soil_density
    global soil_poisson = 0.3
    global soil_bulk = 2.*soil_shear*(1.+soil_poisson)
    soil_bulk = soil_bulk / (3.0*(1.0-2.0*soil_poisson))  
end
[Set_Parameters]
;
block create brick 0 1 0 1 0 [column_height]
block zone generate edgelength 1.0
block zone cmodel assign elastic
block zone property density [soil_density] shear [soil_shear] bulk [soil_bulk]
;
; --- Maxwell Damping Properties forigin 5% Damping ------------
block mechanical damping maxwell (0.0385 0.5) (0.0335 3.5) (0.052 25.0)
;
block gridpoint apply velocity-y 0.0 range position-z -1000 1000
block gridpoint apply velocity-z 0.0 range position-z -1000 1000
block gridpoint apply velocity-x 0.0 range position-z -0.1 0.1
;
; ---- initialize velocit to quarter wave length mode ---
fish define initial_velocity
    local max_vel = 1.0   
    loop foreach local p_gp block.gp.list
        local zcord = block.gp.pos.z(p_gp)
        local theta = (math.pi/2.)*zcord/column_height
        block.gp.vel.x(p_gp) = max_vel*math.sin(theta)
    end_loop
end
[initial_velocity]
;
model history dynamic time-total
block history displacement-x position 0 0 [column_height]
block history velocity-x position 0 0 [column_height]
block history stress-xz position 0 0 0
model solve time-total [num_periods]
;
history export '3' vs '1' table '10'
;
; ------------ function time and amplitude of velocity cycle peaks ----------------
program call 'process.fis'
[data_process('zones.csv')]

benchmark-zones-joints.dat

; Benchmark test forigin Maxwell Damping

; model new
model title 'Maxwell Damping: Vibration of 1D column'
model configure dynamic
model large-strain off
fish automatic-create off
;
fish define Set_Parameters
    ; ------- input parameters ------------------    
    ; global target_frequency = 1  ; frequency in hz
    global num_cycles = 3
    
    global column_height = 100.0
    global height_zones = 100
        
    global soil_density = 2.0;
    local  wave_length = 4.0*column_height
    local  shear_vel = wave_length*target_frequency  
    local  period = 1. / target_frequency
    global num_periods = num_cycles*period
    global soil_shear = shear_vel*shear_vel*soil_density
    global soil_poisson = 0.3
    global soil_bulk = 2.*soil_shear*(1.+soil_poisson)
    soil_bulk = soil_bulk / (3.0*(1.0-2.0*soil_poisson))  

    global bheight = 1.0
    global jks = soil_shear * bheight
    global jkn = jks * 2.0*(1.0+soil_poisson)
    
    ; jointed case, double soil and joint stiffnesses; keep same wave
    soil_shear = 2.0*soil_shear
    soil_bulk = 2.0*soil_bulk
    jkn = 2.0*jkn
    jks = 2.0*jks
 
end
[Set_Parameters]
;
block create brick 0 1 0 1 0 [column_height]
block cut joint-set dip 0 dip-direction 0 spacing [bheight] origin 0 0 0
block create brick 0 1 0 1 -1 0
block zone generate edgelength 1.0
block zone cmodel assign elastic
block zone property density [soil_density] shear [soil_shear] bulk [soil_bulk]
block property density [soil_density]

block insitu stress -1e-6 -1e-6 -1e-6 0 0 0 

block contact jmodel assign mohr
block contact property stiffness-normal [jkn] stiffness-shear [jks] ...
  friction 45 cohesion 1e20 tension 1e20

; --- Maxwell Damping Properties forigin 5% Damping ------------
; zone dynamic damping maxwell (0.0385 0.5) (0.0335 3.5) (0.052 25.0)
block mechanical damping maxwell (0.0385 0.5) (0.0335 3.5) (0.052 25.0)
;
block gridpoint apply velocity-y 0.0 range position-z -1000 1000
block gridpoint apply velocity-z 0.0 range position-z -1000 1000
block gridpoint apply velocity-x 0.0 range position-z -0.1 0.1
; block fix range p-z -1 0
; block fix v-y v-z rot range p-z 0 [column_height]


;
; ---- initialize velocit to quarter wave length mode ---
fish define initial_velocity
    local max_vel = 1.0   
    loop foreach local p_gp block.gp.list
        local zcord = block.gp.pos.z(p_gp)
        local theta = (math.pi/2.)*zcord/column_height
        block.gp.vel.x(p_gp) = max_vel*math.sin(theta)
    end_loop
;    loop foreach local p_b block.list
;        local zcord = block.pos.z(p_b)
;        if zcord > 0
;          local theta = (math.pi/2.)*zcord/column_height
;          block.vel.x(p_b) = max_vel*math.sin(theta)
;        endif
;    end_loop
end
[initial_velocity]
;
model history dynamic time-total
block history displacement-x position 0 0 [column_height]
block history velocity-x position 0 0 [column_height]
block history stress-xz position 0 0 1
block contact history stress-shear-x position 0 0 1

model solve time-total [num_periods]
;
history export '3' vs '1' table '10'
;
;
; --- function time and amplitude of velocity cycle peaks ---
program call 'process.fis'
[data_process('zones-joints.csv')]