Issue 78

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

where ijkl C is the constitutive tensor of the fully solid isotropic material. The global equilibrium condition is expressed in weak form through the residual vector   R U , defined as the difference between the externally applied nodal forces F and the internal force vector derived by integrating the stress field over the undeformed reference volume:   T V dV    R U F B S (4) Equilibrium is achieved when   0  R U . Since Eqn. (4) is nonlinear in U , an incremental-iterative Newton-Raphson procedure is employed to drive the residual to zero. At each iteration, the tangent stiffness matrix T K is assembled as the derivative of the residual with respect to the nodal displacement vector:   0,1 e   is the element density, p is the penalization exponent, and 0 The tangent stiffness matrix accounts for both the material stiffness and the geometric stiffness contributions arising from the current stress state, ensuring quadratic convergence of the iterative scheme in the vicinity of equilibrium. BESO Topology Optimization Framework The topology optimization problem is formulated within the BESO framework, in which material is distributed across the design domain by iteratively assigning each finite element a binary state: solid ( x 1  ) or void (x 0  ). The objective is to minimize the global mean compliance C subject to a prescribed volume fraction constraint f V , yielding the stiffest possible structure for a given material budget. The optimization problem is stated as: T    R U K (5)

T C  U KU

(6)

Minimize:

N

* 1 i V V     i i

(7)



Subject to:

0

*

V

(8)

V  

0

f

V

0

  0,1 , 1,2, , i x i N   

(9)

* V is the target material volume;

where U and K are the global displacement vector and stiffness matrix, respectively;

0 V

is the total design domain volume; f V is the volume fraction; and N is the total number of elements in the design domain. The binary constraint in Eqn. (9) distinguishes BESO from density-based methods and eliminates the need for post-processing to recover a discrete design. i V is the volume of the i -th element;

Sensitivity analysis and filtering The element sensitivity number, which quantifies the contribution of each element to the global compliance, is defined as the elemental strain energy:

1 2

e

T

i i i   u K u i

(10)

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