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Quantum fluctuations are the key concepts of quantum mechanics. Quantum fluctuations of quantum fields induce a zero-point energy shift under spatial boundary conditions. This quantum phenomenon, called the Casimir effect, has been attracting much attention beyond the hierarchy of energy scales, ranging from elementary particle physics to condensed matter physics together with photonics. However, the application of the Casimir effect to spintronics has not yet been investigated enough, particularly to ferrimagnetic thin films, although yttrium iron garnet (YIG) is one of the best platforms for spintronics. Here we fill this gap. Using the lattice field theory, we investigate the Casimir effect induced by quantum fields for magnons in insulating magnets and find that the magnonic Casimir effect can arise not only in antiferromagnets but also in ferrimagnets including YIG thin films. Our result suggests that YIG, the key ingredient of magnon-based spintronics, can serve also as a promising platform for manipulating and utilizing Casimir effects, called Casimir engineering. Microfabrication technology can control the thickness of thin films and realize the manipulation of the magnonic Casimir effect. Thus, we pave the way for magnonic Casimir engineering.