Traditional homogenization formulations for finite volume are extended to allow for the consideration of multiple inclusions and their spatial distributions along with their corresponding individual geometric attributes. For all inclusions present in different states, a combined homogenization formulation (dubbed generalized stiffness formulation, GSF) is posed for summing the five geometric attributes of volume fraction (VF), shape or aspect ratio (AR), orientation, location (position within the domain), and number of inclusions in the domain. For verification, the solutions are compared to two literature-reported calculations of simple cases of domains containing one inclusion of specific shape and orientation in which only one attribute is varied. Once verified for simple cases, GSF stiffness solutions were run for 5040 cases of domains containing multi inclusions with various configuration combinations. These solutions were compared against 36 identical cases of 3-dimensional (3D) domains printed using acrylonitrile butadiene styrene (ABS) and experimentally tested under mechanical compression. Additionally, finite element method (FEM) simulations are run of these test structures under compression. For these test cases, effective composite stiffness values are compared for numerical solutions, experimental tests, and FEM. Relative effects on stiffness of the five inclusion geometric attributes are assessed.

Human joints in general and the human knees particularly are subjected to impacts and wear on a daily basis, specifically during jumping. Considering their inefficient healing and the lack of effective replacements, knee joints became a hot topic for biomechanics research. Multiple works aim to model the knee joint behavior in the literature either experimentally or through finite elements simulations. In this work, we study the effects of the Lebanese folkloric dance Dabke jumping and compare it to vertical jumping. Moreover, we tackle the modeling and simulation of a tibiofemoral knee joint under impact using optimized inverse dynamics and the Brinkman model in the biphasic synovial joint domain. This joint model is mainly made of cartilage, a meniscus (both biphasic low permeability materials), and viscous synovial fluid. Because of the degenerated shape in the thickness direction, classical methods require a large number of degrees of freedom to correctly reproduce the knee behavior. To circumvent this problem, proper generalized decomposition (PGD) model reduction techniques are used to simulate the knee synovial joint. A smart, physically based morphing of space is used to accelerate the simulation process using model reduction techniques. The result of the work is a three-dimensional simulation of the velocity and pressure fields inside the synovial domain, as well as the loads and risks in every muscle and ligament. The experimental results are coupled to the simulations one to derive the main risks involved in both vertical jumping and Dabke jumping.