Issue 78

M. Habashneh et alii, Fracture and Structural Integrity, 78 (2026) 409-431; DOI: 10.3221/IGF-ESIS.78.24

where i K and i u are the stiffness matrix and nodal displacement vector of the i -th element, respectively. Elements with low sensitivity numbers contribute minimally to the structural stiffness and are candidates for removal, while elements with high sensitivity numbers are candidates for reinstatement. Raw sensitivity fields derived from finite element analysis are susceptible to numerical instabilities such as checkerboard patterns and mesh dependency. To mitigate these issues, a nodal sensitivity filtering scheme is applied, in which the sensitivity of each element is replaced by a weighted average computed over a spherical neighborhood of radius min r centered on the element. The filtered sensitivity for the i -th element is given by:



k

n  j

w r

( ) ij



j

1

i 



(11)



k

w r

( ) ij



j

1

j  is the nodal sensitivity of the j -th node, and   ij w r is

where k is the number of nodes within the neighborhood Ω i , n

a linear decay weight function defined as:   ij min ij w r r r  

(12)

in which ij r denotes the distance between the i -th element center and the j -th node. The weight vanishes for nodes outside the filter radius and attains its maximum value at the element center, ensuring that closely neighboring nodes exert the strongest influence on the filtered sensitivity. The nodal sensitivity n j  is itself assembled from the element sensitivities of all M elements connected to node j :

M

1    i

n  j

e  i

(13)

w

i

where the element weight i w reflects the relative spatial proximity of node j to the center of element i :

r

1

ij





w

(1

)

(14)

i



M



M

1

r

ij



i

1

To further stabilize the optimization history and suppress iteration-to-iteration oscillations in the sensitivity field, a two iteration averaging scheme is employed:



z    z

1

i

i

(15)

i 



2

where z denotes the current iteration index. This averaging effectively damps sudden sensitivity reversals that can otherwise cause elements to oscillate between solid and void states across consecutive iterations. The target volume for each successive iteration evolves according to:

1 z z V V ER    (1 )

(16)

where ER is the evolutionary rate governing the pace of material addition or removal per iteration. Once the target volume fraction f V has been reached, the volume is held constant in all subsequent iterations:

*

1 z V V  

(17)

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