Linear Variable Filters (LVFs) are integral to future space missions aimed at measuring the spectral features of distant galaxies. Given the varying red-shifts (shift of light to longer wavelengths as an object moves away from the observer) of these galaxies, an LVF with a resolving power of approximately 10 can effectively identify, classify, and image large populations of galaxies, facilitating studies of cosmic evolution through hyperspectral imaging (HSI) techniques. HSI captures detailed spectral information across a wide range of wavelengths for each pixel in an image, enabling material identification and analysis. Since the galaxies to be studied are very faint, hyperspectral imagers need to be equipped with highly sensitive detectors, cryogenically cooled conditions, and filters with high transmission efficiency. This thesis investigates LVFs based on thin membranes, focusing on materials like silicon nitride (Si3N4) and silicon carbide (SiC). The primary objective is to model these thin and pre-stressed membranes using COMSOL Multiphysics to simulate the thermal expansion and mechanical behaviour of membranes in cryogenic environments. Experimental validation is conducted using Digital Holography Microscopy (DHM) and Laser Doppler Vibrometry (LDV) to ensure the accuracy and consistency of the simulations. The model enables the simulation of membranes with varying dimensions, improving the accuracy of behavioural predictions thereby eliminating the need for physical prototypes for analytical purposes, significantly reducing cost and time. Overall, this research bridges the gap between theoretical models and practical applications of LVFs in space. Enhancing the understanding of material behaviour under extreme conditions advances the development of reliable LVFs for spaceborne optical instruments, offering valuable insights into their durability and performance.